\section{Introduction}

At present, much attention is focused on galaxy clusters
due to the potential application
of the thermal Sunyaev-Zeldovich (SZ) effect as a cosmological tool.
Together with observations of
X-ray emission,
a measurement of the Hubble constant can be made
if a complete sample of galaxy clusters is used
(see reviews by Rephaeli 1995 and Birkinshaw 1998).
Recent advances in
interferometric tools have now allowed accurate mapping
of the SZ decrement, producing two dimensional images which
facilitate comparison
of the X-ray emission and SZ effect.
The SZ effect is typically of
arcminute scale, which is not observable with most
interferometers designed to achieve high angular resolution.
The exception is the Ryle Telescope which has been used successfully to image
the SZ effect at 2 cm.
Another way to achieve the necessary beam size and sensitivity
is to use an interferometer designed for millimeter wavelengths equipped
with low-noise centimeter-wave
 receivers. We used this approach at the Owens Valley Radio
Observatory Millimeter Array (OVRO) and Berkeley-Illinois-Maryland Association Millimeter Array (BIMA), where we have
now detected the SZ effect in over 20 clusters at 1 cm, with
preliminary results given in Carlstrom {\it et al.} (1996, 1997).

The accuracy of cm-wave observations of the SZ effect can be
limited by emission from unresolved
radio point sources towards galaxy clusters.
The observing frequency of 28.5 GHz
was influenced four different factors:
the large beam size required
to be sensitive to the SZ decrement using existing interferometers,
the availability of low-noise HEMT amplifiers, atmospheric transparency,
and the expected low radio source contamination due to falling
flux density
of most radio sources with increase in frequency.
The interferometric technique
makes it possible to detect radio point sources with longer baselines,
which have little sensitivity to the SZ effect,
and then remove their contribution from the short baseline data.
Though such removal will
produce point source-free SZ images, the uncertainties in removal of sources,
due to the limited signal-to-noise and imperfect coherence,
can introduce systematic
noise. When the flux density of point sources are high,
modeling and removal can result in systematic bias levels
comparable to the size of the
SZ effect. Since there are no published surveys
at 28.5 GHz,
it is not possible for us to predict accurately the number of radio sources
expected to be present in a given cluster. In a few hours, however,
 it is possible for us to map a cluster with sufficient
sensitivity to image
unresolved 28.5 GHz radio sources which may complicate SZ mapping.
Clusters with no bright sources,
are then observed for longer periods,
$\sim$ 20 to 50 hours, to obtain adequate signal-to-noise images of the
SZ effect.

In this paper, our primary
goal is to provide information on clusters which contain radio point
sources at 28.5 GHz.
Future
publications will present our results on SZ detections in detail.
 Given that the cluster sample presented in this paper is incomplete
in terms of either redshift or X-ray luminosity,
statistical studies with this sample relating to
cluster properties should be treated with caution.
Section 2 of this paper describes observations made with OVRO and BIMA
arrays. The detected 1 cm radio source sample and its properties are presented
in Section 3, where we also estimate the radio source contamination
in measuring the
Hubble constant through a joint analysis of
SZ and X-ray data by considering
galaxy cluster Abell 2218 as an example.

\section{Observations}

The observed cluster sample is presented in Table 1. The pointing centers of
clusters were derived from existing literature
and were checked with optical images when such images were available.
Usually, optical coordinates of the central galaxy were taken as the
pointing coordinates of a given cluster.
If there was not a clear central galaxy,
centroid coordinates from X-ray observations
were used (e.g., Ebeling {\it et al.} 1996, Ebeling {\it et al.} 1997).
Our sample ranges in redshift
from $\sim$ 0.15 to
0.85, with the lower limit imposed by the large angular
scale of nearby clusters to which the interferometer
would not be sensitive,
and the
upper limit based on the X-ray detection limit of distant clusters.
This sample was observed at OVRO with six telescopes of the
millimeter array during summers of 1995 and 1996,
with six telescopes of the BIMA array during summer of 1996, and
with nine telescopes of the BIMA array during summer of 1997.

We equipped both arrays with low-noise
1 cm receivers, especially designed for the
detection of SZ effect. Each receiver contains a cryogenically cooled
scalar feed-horn and  HEMT
amplifier covering the frequency range 26 to 36 GHz.
The system temperatures scaled above the atmosphere ranged from 30 to 45 K.
During the 1995 OVRO observations,
our receivers were sensitive to linear polarization. Due to
the rotation of polarized
intensity across the sky of our calibrating sources, which
are expected to be polarized up to 10\%,
the calibration process for cluster fields observed in 1995
introduced additional uncertainties.
For long time series calibrator observations with large parallactic angle
coverage, the flux
variation can be corrected by estimating the polarization.
However, for short observations we expect an additional
5\% to 10\% uncertainty in the flux density
of sources imaged in our
1995 cluster sample.
We upgraded our receivers so that observations
during 1996 and 1997 detected circular polarization, which
is not subject to this effect.
For clusters that were initially observed in 1995 and were
reobserved in later years,
we have opted to use latest data to avoid additional uncertainties.

Integration time on
each cluster field ranged from $\sim$ 3 hours to 50 hours,
with the short integration times on clusters where
we happened to detect a bright radio source.
For each cluster, $\sim$ 5 minute
observations of a secondary calibrator from the VLA
calibrator list were interleaved
with every $\sim$ 25 minutes spent on a cluster.
Between different clusters, $\sim$ 45 to 60 minutes
were spent observing planets, with
care taking to observe
Mars frequently since it is used as our primary flux calibrator.
The flux densities of the secondary gain and phase calibrators were
calibrated relative to Mars.
The brightness temperature of Mars was
calculated
using a thermal-radiative model with an estimated uncertainty
of 4\% (Rudy 1987). In Table 2, we present 1 cm flux densities of gain
and phase calibrators determined
through this process for the summer 1997 observations, which can
be useful for future observational programs at this wavelength.
Some of these calibrator sources are likely to be variable at
28.5 GHz,
but during the time scale of our 1997 observations, 2 months,
the maximum variation was found to be less than 4\%.
The uncertainties in the reported
flux densities in Table 2 are less than $\sim$ 5\%.

For the OVRO data, the MMA software package (Scoville {\it et al.} 1993)
was used to calibrate the visibility data and then write it in UV-FITS
 format. We flagged all of the data taken when one
antenna was shadowed by another, cluster data that was not bracketed in time
by phase calibrator data (mostly at the end or beginning of an
observation), and, rarely, data with anomalous correlations.
 We followed the same procedure for data from BIMA,
 except that MIRIAD software package (Wright \& Sault 1993)
was used for calibration and data editing
purposes. The image processing and CLEANing were done using
DIFMAP (Shepherd, Pearson, \& Taylor 1994). We cleaned all fields
uniformly, based on the rms noise level.
Our automated
mapping algorithm within DIFMAP was able to find sources with
extended structures, which when compared with low frequency
data, such as VLA D-Array 1.4 GHz NVSS survey (Condon {\it et al.} 1996),
were confirmed for all
cases. In general, $\sim$ 2000 clean iterations
with a low clean loop gain of 0.01 was chosen to avoid instabilities
and artifacts that can occur in fields with a large number of sources.

We looked for radio point sources in naturally weighted maps
with visibilities
greater than 1 k$\lambda$ in BIMA data and 1.5 k$\lambda$ in OVRO data.
Since the interferometer is less sensitive with only the long baseline data,
we obtained flux densities of detected sources in maps made with
all visibilities. Using images made with all the UV data
also allowed us to look for
sources with extended structure.
Typical synthesized beam sizes in these images
range from 12$''$ to 30$''$.
For typical cluster and control blank fields with
no bright radio sources ($\geq$ 1 mJy), and no evident SZ decrement,
the noise
distribution was found to be a Gaussian centered at zero.
These images did not contain any pixels with peak flux density
$\leq$ -4 $\sigma$ within 250$''$.
The mean rms noise level for all our
56 cluster observations is
0.24 mJy beam$^{-1}$,
while the lowest rms noise level is 0.11 mJy beam$^{-1}$ for
BIMA observations and 0.07 mJy beam$^{-1}$ for OVRO observations.
Given the decrease in sensitivity due to the
primary beam attenuation from the image centers,
we only report sources within 250$''$ of the pointed coordinates.
A Gaussian-noise analysis suggested that within 250$''$ from the center
in all 56 cluster fields, only 1 noise pixel is expected
at a level above 4 $\sigma$.
Among all
56 cluster fields, there was only
one instance where a source was
clearly detected at a distance greater
than 250$''$ from the cluster center;
In CL 0016+16 we found a source $\sim$ 290$''$ away from the
pointed coordinates, which is
discussed in Carlstrom {\it et al.} (1996).

\section{Results and Discussion}

In Table 3, we report the flux densities of detected 1 cm radio sources.
When calculating these flux densities, we have corrected for the
beam response.
To determine the primary beam pattern at BIMA, the radio source 3C454.3 with
a flux density of $\sim$ 8.7 Jy at 28.5 GHz was
observed with a grid pattern of pointing offsets, and then a two dimensional
Gaussian fit was performed to the flux density values.
A 300$''$ by 300$''$ grid with 75$''$ spacing was
best fit by a Gaussian with a major
axis of 386$''$, a minor axis of 380$''$ (FWHM)
and a position angle -85.31$^{\circ}$, with an uncertainty of 3$''$.  The
rms residual from the fit was $\sim$ 0.01 Jy.  A 360$''$ by 360$''$
grid with 90$''$ spacing was best fit by a Gaussian with a
major axis of 382$''$ and
a minor axis of 379$''$, also with rms residual of 0.01 Jy.
 Given the small
difference between two axes and positional uncertainty of at least
$\sim$ 5$''$ at BIMA, we have utilized a symmetrical Gaussian model with a
380$''$ FWHM half-power point.
At OVRO, we have made holographic measurements of
the beam pattern and have corrected the fluxes based on the position of sources
relative to a modeled Gaussian distribution, which resulted
in a primary beam of 235$''$ (FWHM).

For our 1 cm sample,
we searched literature for low
frequency counterparts within 15$''$ from the 28.5 GHz radio source coordinates.
A low frequency source was accepted
as a counterpart when the difference between our coordinates
and published coordinates was less than
the astrometric uncertainty
in our coordinates and the low frequency counterpart coordinates. The error
in 1 cm coordinates ranges from $\sim$ 3$''$ to 10$''$, which is equivalent
to  the
image resolution divided by the signal-to-noise with which
the source was detected.
For cluster fields with bright radio sources,
the signal-to-noise was low due to small
integration times, producing uncertainties in position as high as $\sim$
10$''$.
Still, identification of such sources was easier due to their
relatively high flux densities.
For published sources, the astrometric errors ranged from sub-arcseconds,
mostly from VLA observations, to few arcseconds. The mean
difference between our coordinates and published coordinates was
$\sim$ 6$''$. Based on Moffet \& Birkinshaw (1989), we estimated the
field density of 5 GHz radio sources towards clusters with a flux
limit of 1 mJy
is $\sim$ 25 degree$^{-2}$. Therefore, the
probability of an unrelated radio source, with a 5 GHz flux density above
1 mJy, lying within 6$''$ is $<$ 0.5\%.

When there is a well known
counterpart from literature coincident with the
detected 1 cm source, we have noted the commonly used name
in Table 3.
We have
calculated the spectral index
of individual radio sources by fitting all known
flux densities,
with spectral index $\alpha$ defined as
$S \sim \nu^{-\alpha}$. In Fig.\ 1, we show a histogram of the calculated
 spectral indices of 52 sources for which we have
found radio observations at
 other frequencies. In this plot, we have
not included the sources 1635+6613 towards Abell 2218 and 1615-0608 towards
Abell 2163,
which are found with flux densities that peak between
1.4 GHz and 28.5 GHz (e.g., Fig.\ 2).
These sources may indicate self-absorbed
radio cores, with spectral turnover due to free-free absorption.
Such turnovers in inverted spectra are found in Gigahertz Peaked Spectrum (GPS)
sources, though definition of GPS sources calls for peaked spectra between
0.5 and 10 GHz (De Vries, Barthel, \&  O'Dea 1997).
The increase in turnover frequency well above
10 GHz, could be due to an increase in ambient density.
Also, 1615-0608 towards Abell 2163 is known to be variable based on VLA
observations by Herbig \& Birkinshaw (1994). During our observations, the flux
density of this source did not change significantly:
we measured a flux density of 1.12 $\pm$ 0.29 mJy in 1995 (OVRO) and
0.93 $\pm$ 0.42 mJy in 1997 (BIMA) at 1 cm.
In Table 3, we report the 1995 flux density value since
tabulated VLA measurements were made closer to our 1995 observations. Abell 2163 is also known to contain one of the largest radio halo sources ever found.
We did not detect any emission from the cluster center, which is
understandable given that the halo was detected only at 1.4 GHz, with
an integrated flux density of $\sim$ 6 mJy and a steep spectral
index of $\sim$ 1.5.

In our sample we also find 3 sources with inverted spectra between
1.4 and 28.5 GHz:
0152+0102 towards Abell 267, 0952+5151 towards Zw 2701,
and 1155+2326 towards Abell 1413.
These could
either represent free-free emission due to starburst, or synchrotron emission
from a weak AGN, or both, with an optically thick part of a thermal
bremsstrahlung component that extends to high frequencies. Since the
inverted spectral indices are less than -2, which is the value expected for
optically thick thermal sources, it is more likely that these sources
represent multiple non-thermal components.
The relatively flat-spectrum ($-0.5 \lesssim
\alpha \lesssim 0.5$) sources may indicate unresolved cores and hotspots,
and further
high resolution observations are necessary to resolve full
structure. These sources include 0152+0102 towards Abell 267 and 2201+2054 towards Abell 2409.

In Table 3, the identification
 of a source as a central galaxy (CG) was only made when we have
used the optical coordinates of central galaxy
from literature as the pointing coordinates,
and when we have detected a radio source at 1 cm within 10$''$ of the observed
coordinates. We have found 13 such
 sources, which
may well represent the radio emission associated with
central
cD galaxy of the cluster.

Due to the low resolution of our observations,
most of the radio sources are unresolved, but in a few cases
we find some evidence for extended emission. These sources include
0037+0907 and 0307+0908 towards Abell 68 (Fig.\ 3), 1716+6708 (4C +67.26)
towards RXJ1716+6708 (Fig.\ 4), 1335+4100 (4C +41.26)
towards Abell 1763, and 1017+5934  towards Abell 959.
The nature of extended emission associated with these sources
should be further studied, and high resolution observations at several
frequencies will be helpful in this regard.  1335+4100 (4C +41.26)
towards Abell 1763 is a well studied FR II type radio source (e.g., Owen
1975).

For our sample of 52 radio sources with known flux densities at lower
frequencies,
a mean spectral index of 0.77 $\pm$ 0.06, and a median of 0.84
are found. If the three sources with
inverted spectral indices are not considered, the mean and median rise to
0.85 $\pm$ 0.06 and 0.85 respectively.
To avoid a biased estimate for the spectral index distribution,
however, we must consider counterparts at 1 cm for all
sources detected at lower frequencies. Galaxy clusters
CL 0016+16, Abell 2218 and Abell 665 have been observed
at 1.4, 4.85, 14.9 and 20.3 GHz by Moffet \& Birkinshaw (1989),
and their observations are complete to a flux density limit of
1 mJy at 4.85 GHz.
In each of these three clusters, we selected sources in the low frequency
survey which were located within 300$''$ from the cluster center.
We list these sources, their flux densities at 1.4 GHz,
expected flux densities at 28.5 GHz based on 1.4 and 4.85
GHz spectral index, observed flux densities at 28.5 GHz, and calculated
spectral indices between 1.4 and 28.5 GHz in Table 4.
At 28.5 GHz, we detect all sources with flux densities greater than 7 mJy
at 1.4 GHz, at a detection level greater than
3 $\sigma$. We looked for counterparts of these sources at 28.5 GHz,
which should form
a complete sample and not bias the spectral index distribution.
Also, given that we looked for 28.5 GHz
counterparts only within 15$''$ of the 1.4 GHz source coordinates,
we expect all detections at a level above 3$\sigma$ to be real.
For this sample, we find a mean spectral index of
0.71 $\pm$ 0.08, and a median of 0.71.
The 1 cm spectral
index distribution agrees
with that of the 6 cm mJy population with a median of 0.75
(Donnelly {\it et al.} 1987).
However, the 1 cm distribution is steeper than the
sub-mJy and
the $\mu$Jy populations, where medians of 0.35
(Windhorst {\it et al.} 1993) and
0.38 (Fomalont {\it et al.} 1991) were found at 4.85 and 8.4 GHz respectively.
The latter sub-mJy populations have been identified with
faint blue galaxies.
Our sample could be part of the lower frequency
mJy and sub-mJy
populations, but given the lack of detailed optical
data for most of our sources,
we cannot exactly state the optical nature of our 28.5 GHz sample.

We compare our results with a 1.4 GHz survey by Condon, Dickey, \& Salpeter
(1990) in areas without rich galaxy clusters, in
order to address whether we are finding an overabundance of radio
sources at 28.5 GHz towards galaxy clusters.
They found a total of 354 radio sources, down to a flux
limit of 1.5 mJy, in a total surveyed area of about 12 square degrees.
Seven of
these sources are thought to be associated with galaxy clusters, which
includes
 Abell 851 (source 0942+4658 in Table 3). Ignoring this small
contamination, we calculated the expected flux densities
of the 1.4 GHz sources at 28.5 GHz based on the mean
spectral index value of our sample.
For a spectral index of  0.71, we found 170 sources with fluxes greater than
0.4 mJy at 1 cm,
which is the lowest 4 $\sigma$ detection limit of our observations.
Given that the ratio of total
area observed
by the 1.4 GHz survey and our survey is $\sim$ 15, we only expect $\sim$ 7 to
15 sources be present with flux densities greater than
0.4 mJy at 28.5 GHz, and therefore be detected in our observations.
Given that we find 62 sources, ignoring
the inverted and unusual spectrum sources, we conclude that we are finding
at least $\sim$ 4 times more sources than usually expected.
Given the primary beam attenuation and the nonuniformity
of flux density limit from one cluster field to another, the above ratio is
only a lower bound on the calculated ratio.
If we take these facts into account, we find that
our sample at 28.5 GHz contains 7 times
more than one would normally
expect based on a low frequency
radio survey devoid of clusters. This result may have some consequences
when planning and reducing data from large field observations, such as
our planned degree-square SZ effect survey.

The reason that we are seeing more sources might be
explained
through gravitational lensing of a background
radio population by cluster potentials.
Our cluster sample ranges in redshift from $\sim$ 0.15 to 0.85, with a
mean of 0.29 $\pm$ 0.02, and a median of 0.23.
If the excess source counts are indeed an effect due to lensing,
the background population should be at a lower flux level than what
we have observed. An optimal lensing
configuration suggests that background
sources should be at angular diameter distances twice that of the galaxy
clusters, which are assumed to be lensing potentials.
For the range of cluster redshifts, the background source
sample should be at redshifts between $\sim$
0.4 and 1.4, with a mean redshift of
$\sim$ 0.7. In terms of well known radio source samples, a
possibility of such an unlensed population between redshifts of 0.4 and 1.4 is
the sub-mJy radio sources at 5 and 8.4 GHz (Windhorst {\it et al.}
1993).
For simplicity, we consider a
cluster potential based on the singular isothermal sphere (SIS) model of
Schneider, Ehlers \& Falco (1992). Such a potential brightens, but dilutes the spatial distribution,
background
sources by the magnification factor,
\begin{equation}
\mu(\theta) = \left| 1 - \frac{\theta_E}{\theta} \right|^{-1},
\end{equation}
where $\theta$ is the angle, or
distance, to radio source from cluster center,
and $\theta_E$ is the
Einstein angle, which depends on the distances to a given cluster and
background radio sources ($\theta > \theta_E$).
The Einstein angle can be observationally determined through
optical images of clusters where background sources are lensed
into arcs and whose redshift is known. In order to estimate
reliable Einstein angles for background sources
at redshifts around $\sim$ 0.7,
we considered two well studied clusters. In
Abell 2218 an arc is found $\sim$ 21$''$ from the cluster center
with a measured redshift of 0.702 (Pell\'{o} {\it et al.} 1992), and
in Abell 370 an arc is found $\sim$ 36$''$, with a redshift of 0.72 (Kneib
{\it et al.} 1994).
The 1 cm source sample ranges from $\sim$ 0 to 250$''$ in distance from
individual cluster centers, with a mean of $\sim$ 94 $\pm$ 10$''$, and
a median of 97$''$.
These values suggest a mean magnification factor of $\sim$ 1.4, suggesting
that we should expect 10 to 20 sources at 28.5 GHz towards
galaxy clusters, based on our earlier estimate of 7 to 15 sources and not
accounting for the spatial dilution due to lensing.
It is unlikely
that lensing can account for the significant excess number of radio sources
we have detected at 28.5 GHz.
Also, VLA A-array observations at 1.4 GHz to a flux density
limit of 1 mJy
have not yet produced convincing evidence for the existence of
gravitationally lensed radio sources, such as radio arcs,
towards galaxy clusters (e.g. Andernach {\it et al.} 1997).
An apparent detection of gravitational lensing towards clusters, based on the
tangential orientation of radio sources, is discussed in
Bagchi \& Kapahi (1995). Recently, Smail {\it et al.} (1997) have
observed an increase in sub-mm surface flux density towards clusters Abell
370 and CL 2244-02, which was interpreted as due to the gravitational lensing
by cluster potentials of
strongly star-forming galaxies at redshifts $\gtrsim$ 1.
Given that the number counts of our sample cannot be totally explained
as due to a lensing effect and that
we do not have enough resolution to look for alterations that might
be a result of lensing situations, we conclude that
a large fraction of the detected 1 cm  sample
must be associated with clusters towards which they were found.

In Fig.\ 6, we plot the source counts per solid angle
at 1 cm, which were binned in logarithmic intervals of 0.2 mJy into
8 different bins. The solid angle for each flux bin takes into account the
variation in sensitivity of our observations.
A large number of sources are found in the lowest bin,
which may have a nonuniform detections
due to
the variation in noise level from one cluster field to another.
The maximum-likelihood
fit to a power-law distribution of the observed sources,
and normalised to the
source counts greater than 1.6 mJy,
is $N(>S) = (59 ^{+20}_{-15}) \times (S/$mJy$)^{-0.96 \pm 0.14}$
in a total surveyed area of 2.5 $\times$ 10$^{-4}$ sr,
where  $N(>S)$ is an integral representation of number of
sources with flux densities greater than $S$ in mJy.
Given
that we only looked for sources towards a
sample of X-ray luminosity selected galaxy clusters, and that we have
not carried out 28.5 GHz observations to a uniform flux density limit in all
observed fields, the above number count-flux  relationship
should not be treated
as true in general for all radio sources at 28.5 GHz.
However, our result may be useful when studying radio source contamination
in planned CMB anisotropy and  SZ experiments.
The corresponding differential source count slope $\gamma$ is  $\sim$
1.96 ($dN/dS \propto S^{-\gamma}$).
This slope
 is similar to what is found for 6 cm mJy radio sources ($\gamma$ $\approx$
1.8, Donnelly {\it et al.} 1987)
with similar flux densities as our sample, but is marginally flatter than
sub-mJy population of radio sources ($\gamma$ $\approx$ 2.3,
Windhorst {\it et al.} 1993).
The flattening of the slope from the expected Euclidean value ($\gamma = 2.5$)
is likely to be due to the dependence of radio luminosity
with galaxy cluster properties, as we may be finding
bright radio sources towards X-ray luminous clusters.

Recently, Loeb \& Refregier (1997) have suggested that
the value of the
Hubble constant determined through a joint
analysis of SZ and X-ray data may be underestimated due to radio
point source contamination.
We address this issue based on our 28.5 GHz data and low
frequency observations towards Abell 2218, which was also
studied
in Loeb \& Refregier (1997).
There are 5 known sources within 300$''$ from the cluster
center (Moffet \& Birkinshaw 1989),
out of which we detect 3 (see Fig.\ 5) down to a flux density of
$\sim$ 1 mJy.
By subtracting these three sources and using all visibilities
and a Gaussian UV taper of 0.5
at 0.9 k$\lambda$, we find a SZ decrement
with a signal-to-noise ratio greater than 20. The restored beam size of
this map is 110$''$ by 98$''$.
By extrapolating low frequency flux densities to 1 cm, based on the
1.4 and 4.85 GHz spectral indices, we infer an unaccounted intensity
of $\sim$ 250 Jy sr$^{-1}$. Assuming that the
flux densities in the map at the expected location of the unsubtracted
sources are the real 28.5 GHz flux densities of the undetected
sources, we estimate an upper limit on
the unaccounted intensity of $\sim$ 590 Jy sr$^{-1}$.
The latter value is equivalent to the noise contribution in the
observed SZ decrement.
The two intensities are equivalent to $\sim$ 10 and 25 $\mu$K respectively,
which we take as the range of errors in the observed
SZ temperature decrement $\Delta T_{sz}$.
The central temperature fluctuation
due to SZ decrement towards Abell 2218, $\Delta T_{sz}$
ranges from $\sim$ 0.6 to 1.1 mK, based on different $\beta$-models
to SZ morphology (see also Jones {\it et al.} 1993). The
Hubble constant, $H$, varies as $H \propto {T_{sz}}^{-2}$.
Thus, the offset in true and calculated Hubble constant, $\Delta H$, is:
\begin{equation}
\frac{\Delta H}{H} \sim \frac{2 \Delta T_{sz}}{T_{sz}}.
\end{equation}
For Abell 2218, we find that the fractional correction
to the Hubble constant from not accounting for sources
with flux densities less than 0.5 mJy at 28.5 GHz ranges from
$\sim$ 1\% to 6\%.
If sources with flux densities less than 0.1 mJy are not accounted for,
we estimate an upper limit on the
offset of 2\%.
These values are in agreement with Loeb \& Refregier (1997), who suggested
that the 5 GHz
sub-mJy population (Windhorst {\it et al.} 1993)
may affect the derivation of the Hubble constant at 15 GHz by 7\% to 13\%, if
sources less than 0.1 mJy at 15 GHz not properly taking into account.
Given that the intensity of the SZ decrement
has a spectral index of -2, and assuming a spectral
index of 0.7 for the radio source flux contribution,
we estimate the frequency dependence
 of the correction as $\nu^{-2.7}$.
Thus, at 15 GHz, we also find that the Hubble constant may be
underestimated up to 13\%. The contribution from free-free emission, which
scales as $\nu^{-0.1}$, is not expected to contribute to
underestimation of the Hubble constant
at a level more than 0.1\% at 28.5 GHz. At high frequencies ($>$ 90 GHz),
the free-free and dust emissions, with dust scaling as
$\nu^{\beta}$, $3<\beta<4$ at 100 GHz, may become the dominant source of error.
Therefore, based on the 28.5 GHz data towards Abell 2218,
we conclude that the error in the Hubble constant through a
joint analysis of SZ data at 28.5 GHz and X-ray emission observations
 is not expected to be larger than the error introduced by the analysis
(such as $\beta$-models)
and unknown nature of the galaxy cluster shape (oblate vs. prolate etc.), which
can amount up to 30\% (e.g. Roettiger {\it et al.} 1997).

\acknowledgments
We wish to thank the staff at OVRO and BIMA observatories for
their assistance with our observations, in particular
J. R. Forster,
J. Lugten, S. Padin,
R. Plambeck, S. Scott,  and D. Woody. We also thank C. Bankston and P.
Whitehouse at the MSFC for helping with the construction of the SZ receivers,
and
M. Pospieszalski for the Ka-band HEMT amplifiers. We also gratefully
acknowledge H. Ebeling, A. Edge, H. Bohringer, S. Allen, C. Crawford,
A. Fabian, W. Voges, J. Huchra and P. Henry for providing us results from
X-ray observations of galaxy clusters prior to publication. ARC acknowledges
useful discussions with A. Fletcher on an early draft of the
paper. JEC acknowledges
support from a NSF-Young Investigator Award and the David and Lucile
Packard Foundation. Initial support to build
hardware for the SZ observations came from a NASA CDDF grant.
Radio astronomy with the OVRO millimeter array is supported
by the NSF grant AST96-13717, and astronomy with the BIMA array is
supported by the NSF grant AST96-13998.


\section{Introduction}

A celebrated theorem by Dirac \cite{Dirac52} asserts the existence of a Hamilton cycle whenever the minimum degree of a graph $G$, denoted $\delta(G)$, is at least $\frac{n}{2}$. Moreover, this is best possible as can be seen from the complete bipartite graph $K_{\floor{\frac{n-1}{2}},\ceil{\frac{n+1}{2}}}$. Dirac's theorem is one of the most influential results in the study of Hamiltonicity of graphs and has seen generalisations in many directions over the years (for some examples consider surveys \cite{lisurvey,gouldsurvey,bennysurvey} and references therein). In this paper we discuss one such direction by considering what conditions ensure that we can find various \textit{$2$-factors} in $G$. Here, a \textit{$2$-factor} is a spanning 2-regular subgraph of $G$ or equivalently, a union of vertex-disjoint cycles that contains every vertex of $G$ and hence, $2$-factors can be seen as a natural generalisation of Hamilton cycles. Brandt, Chen, Faudree, Gould and Lesniak \cite{brandt97} proved that for a large enough graph the same degree condition as in Dirac's theorem, $\delta(G)\ge n/2$, allows one to find a $2$-factor with exactly $k$ cycles.

\begin{thm}
If $k \geq 1$ is an integer and $G$ is a graph of order $n \geq 4k$ such that $\delta(G) \geq \frac{n}{2}$, then $G$ has a $2$-factor consisting of exactly $k$ cycles.
\end{thm}

Once again, this theorem gives the best possible bound on the minimum degree, using the same example as for the tightness of Dirac's theorem above. This indicates that perhaps if we restrict our attention to Hamiltonian graphs, thereby excluding this example, a smaller minimum degree might be enough. That this is in fact the case was conjectured by Faudree, Gould, Jacobson, Lesniak and Saito \cite{Faudree05}.

\begin{conj}\label{conj:main}
For any $k \in \mathbb{N}$ there are constants $c_k <1/2,$ $n_k$ and $a_k$ such that any Hamiltonian graph $G$ of order $n\ge n_k$ with $\delta(G) \ge c_k n+a_k$ contains a $2$-factor consisting of $k$ cycles.
\end{conj}

Faudree et al.\ prove their conjecture for $k=2$ with $c_2=5/12.$

The conjecture was shown to be true for all $k$ by S\'{a}rk\"{o}zy \cite{Sarkozy08} with $c_k=1/2-\varepsilon$ for an uncomputed small value of $\varepsilon>0.$ Gy\"ori and Li \cite{gyori12} announced that they can show that $c_k=5/11+\varepsilon$ suffices. The best known bound was due to DeBiasio, Ferrara and Morris \cite{DeBiasio14} who show that  $c_k= \frac{2}{5}+\varepsilon$ suffices.

On the other hand no constructions of very high degree Hamiltonian graphs without $2$-factors of $k$ cycles are known. Faudree et al.~ \cite{Faudree05} say ``we do not know whether a linear bound of minimum degree in Conjecture~\ref{conj:main} is appropriate''. Sark\"ozy~\cite{Sarkozy08} says ``the obtained bound on the minimum degree is probably far from
best possible; in fact, the ``right'' bound might not even be linear''. DeBiasio et al.~\cite{DeBiasio14} say ``one vexing aspect of Conjecture~\ref{conj:main} and the related work described here is that it is possible that a sublinear, or even constant, minimum degree would suffice to ensure a Hamiltonian graph has a 2-factor of the desired type''. In particular, in \cite{Faudree05,Sarkozy08,DeBiasio14} they all ask the question of whether the minimum degree needs to be linear in order to guarantee a $2$-factor consisting of $k$ cycles. We answer this question by showing that the minimum degree required to find $2$-factors consisting of $k$ cycles in Hamiltonian graphs is indeed sublinear in $n.$

\begin{thm}\label{thm:main}
For every $k\in \mathbb{N}$ and $ \varepsilon>0$, there exists $N = N(k,\varepsilon)$ such that if $G$ is a Hamiltonian graph on $n \geq N$ vertices with $\delta (G) \geq \varepsilon n$, then $G$ has a $2$-factor consisting of $k$ cycles.
\end{thm}

\subsection{An overview of the proof}
We now give an overview of the proof to help the reader navigate the rest of the paper.

In the next section we will show that any $2$-edge-coloured graph $G$ on $n$ vertices with minimum degree being linear in both colours contains a blow-up of a short colour-alternating cycle. This is an auxiliary result which we need for our main proof. There, we also introduce ordered graphs and show a result which, given an ordering of the vertices of $G$ allows us to find a blow-up as above that is also consistent with the ordering, meaning that given two parts of the blow-up, vertices of one part all come before the other.

The main part of the proof appears in \Cref{sec:main}. The key idea is given a graph $G$ with a Hamilton cycle $H=v_1\ldots v_nv_1$, to build an auxiliary $2$-edge-coloured graph $A$ whose vertex set is the set of edges $e_i=v_iv_{i+1}$ of $H$ and for any edge $v_iv_j\in G\setminus H$ we have a red edge between $e_i$ and $e_j$ and a blue edge between $e_{i-1}$ and $e_{j-1}$ in $A$.
The crucial property of $A$ is that given any vertex disjoint union of colour-alternating cycles $S$ in $A$ one can find a $2$-factor $F(S)$ in $G$, consisting of the edges of $H$ which are not vertices of $S$ and the edges of $G$ not in $H$ which gave rise to the edges of $S$ in $A$.

However, we can not control the number of cycles in $F(S)$ (except knowing that $F(S)$ has at most $|S|$ cycles), since it depends on the structure of $S$ and also on how $S$ is embedded within $A$. To circumvent this issue we will find instead a large blow-up of $S$. Then within this blow-up we show how to find a modification of $S$ denoted $S^+$ which has the property that $F(S^+)$ has precisely one cycle more than $F(S)$. Similarly, we find another modification $S^-$ such that the corresponding $2$-factor $F(S^-)$ has precisely one cycle less than $F(S)$. Since the number of cycles in $F(S)$ is bounded, if our blow-up of $S$ is sufficiently large we can perform these operations multiple times and therefore obtain a $2$-factor with the target number of cycles.

\section{Preliminaries}\label{sec:prelim}
Let us first fix some notation and conventions that we use throughout the paper. For a graph $G=(V,E)$, let $\delta(G)$ denote its minimum degree, $\Delta(G)$ its maximum degree and $d(v)$ the degree of a vertex $v \in V$. For us, a \emph{$2$-edge-coloured graph} is a triple $G = (V, E_1, E_2)$ such that both $G_1 = (V, E_1)$ and $G_2 = (V,E_2)$ are simple graphs. We always think of $E_1$ as the set of \emph{red} edges and of $E_2$ as the set of \emph{blue} edges of $G$. Accordingly, we define $\delta_1(G)$ to be the minimum degree of red edges of $G$ (that is $\delta(G_1)$), and analogously $\Delta_1(G)$, $\delta_2(G)$, etc. Note that with our definition the same two vertices may be connected by two edges with different colours. In this case, we say that $G$ has a \emph{double edge}. A \textit{blow-up} $G(t)$ of a $2$-edge coloured graph $G$ (with no two vertices joined by both a red and a blue edge) is constructed by replacing each vertex $v$ with a set of $t$ independent vertices and adding a complete bipartite graph between any two such sets corresponding to adjacent vertices in the colour of their edge. When working with \emph{digraphs} we always assume they are simple, so without loops and with at most one edge from any vertex to another (but we allow edges in both directions between the same two vertices).

\subsection{Colour-alternating cycles}\label{subs:cycles}
In this subsection, our goal is to prove that any $2$-edge-coloured graph, which is dense in both colours contains a blow-up of a colour-alternating cycle. We begin with the following auxiliary lemma that will only be used in the subsequent lemma where we will apply it to a suitable auxiliary digraph to give rise to many colour-alternating cycles.
\begin{lem}
Let $k \ge 2$ be a positive integer. A directed graph on $n$ vertices with minimum out-degree at least $\frac{n\log (2k)}{k-1}$ has at least $\frac{n^\ell}{2k^{\ell+1}}$ cycles of length $\ell$ for some fixed $2\le \ell\le k.$
\end{lem}

\begin{proof}
Let us sample $k$ vertices $v_1,\ldots, v_{k}$ from $V(G),$ independently, uniformly at random, with repetition. We denote by $X_i$ the event that vertex $v_i$ has no out-neighbour in $S:=\{v_1,\ldots, v_k\}.$ We know that $\mathbb{P}(X_i)\le \left(1-\frac{\log (2k)}{k-1}\right)^{k-1}\le \frac{1}{2k}.$ If no $X_i$ occurs then the subgraph induced by $S$ has minimum out-degree at least $1$ so contains a directed cycle. The probability of this occurring is at least:
$$ \mathbb{P}\left(\overline{X_1}\cap \ldots \cap \overline{X_k}\right)=1-\mathbb{P}(X_1\cup \ldots \cup X_k)\ge 1-k\mathbb{P}(X_i) \ge 1/2,$$
where we used the union bound. This means that in at least $n^k/2$ outcomes we can find a cycle of length at most $k$ within $S.$ In particular, there is an $\ell \le k$ such that in at least $\frac{n^k}{2k}$ outcomes the cycle we find has length exactly $\ell$. Note that the same cycle might have been counted multiple times, but at most $k^\ell n^{k-\ell}$ times. This implies that $C_\ell$ occurs at least $\frac{n^\ell}{2k^{\ell+1}}$ times.
\end{proof}

Now, we use this lemma to conclude that there are many copies of some short colour-alternating cycle in any $2$-edge-coloured graph which has big minimum degree in both colours.

\begin{lem}\label{lem:many-cycles}
For every $\gamma \in (0,1)$ there exist $c=c(\gamma), L = L(\gamma)$ and  $K = K(\gamma)$ such that, if $G$ is a $2$-edge-coloured graph on $n \geq K$ vertices satisfying $\delta_1(G), \delta_2(G) \geq \gamma n$, then $G$ contains at least $cn^\ell$ copies of a colour-alternating cycle of some fixed length $4 \le \ell \le L$.
\end{lem}
\begin{proof}
Let $k=8/\gamma^2 \log (8/\gamma^2)$ so that $\gamma^2/4 \ge \log (2k)/(k-1).$ We set $L=2k,$ $K=8k/\gamma^2$ and $c=(\gamma/2)^{2k}/(4k^{k+1}).$
We build a digraph $D$ on the same vertex set as $G$ by placing an edge from $v$ to $u$ if and only if there are at least $\gamma^2n/2$ vertices $w$ such that $vw$ is red and $wu$ is blue.

Let us first show that every vertex of $D$ has out-degree at least $\gamma^2n/4.$ There are at least $\gamma n$ red neighbours of $v$ and each has $\gamma n$ blue neighbours so there are at least $\gamma^2n^2$ red-blue paths of length $2$ starting at $v.$ Let us assume that there are less than $\gamma^2n/2$ vertices $u$ such that there are at least $\gamma^2n/2$ vertices $w$ such that $vw$ is red and $wu$ is blue.
In this case there are less than $\gamma^2n/2 \cdot n+ n \cdot \gamma^2n/2$ red-blue paths starting at $v$ which is a contradiction. Note that we allowed $u=v$ in the above consideration so we deduce that minimum out-degree in $D$ is at least $\gamma^2n/2-1\ge \gamma^2n/4.$
The previous lemma implies that there is some $\ell \le k$ such that $D$ contains at least $n^{\ell}/(2k^{\ell+1})$ copies of $C_\ell.$

For any such cycle by replacing each directed edge by a red-blue path of $G$ between its endpoints, ensuring we don't reuse a vertex, we obtain at least $(\gamma^2n/2-\ell)(\gamma^2n/2-\ell-1)\cdots(\gamma^2n/2-2\ell+1)\ge (\gamma/2)^{2\ell}n^\ell$
colour-alternating $C_{2\ell}$'s in $G$. Noticing that each such $C_{2\ell}$ may arise in at most $2$ different ways from a directed $C_{\ell}$ of $D$ we deduce that there are at least $n^{\ell}/(2k^{\ell+1})\cdot(\gamma/2)^{2\ell}n^\ell/2\ge c(\gamma)n^{2\ell}$ colour-alternating $C_{2\ell}$'s in $G$.
\end{proof}

The reason for formulating the above lemma is that we can deduce the existence of the blow-up of a cycle from the existence of many copies of this cycle using the hypergraph version of the celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem proved by Erd\H{o}s in \cite{kst}:

\begin{thm}\label{thm:kst}
Let $\ell,t \in \mathbb{N}$. There exists $C=C(\ell,t)$ such that any $\ell$-graph on $n$ vertices with at least $Cn^{\ell-1/t^\ell}$ edges contains $K^{(\ell)}(t)$, the complete $\ell$-partite hypergraph with parts of size $t,$ as a subgraph.
\end{thm}

We are now ready to find our desired blow-up.
\begin{lem}\label{lem:cycle-blow-up}
For every $\gamma \in (0,1)$ and $t\in \mathbb{N}$, there exist positive integers $L = L(\gamma)$ and  $K = K(\gamma,t)$ such that, if $G$ is a $2$-edge-coloured graph on $n \geq K$ vertices satisfying $\delta_1(G), \delta_2(G) \geq \gamma n$, then $G$ contains $\mathcal{C}(t6^L)$ where $\mathcal{C}$ is a colour-alternating cycle with $|V(\mathcal{C})| \leq L.$
\end{lem}
\begin{proof}
Let $L=L(\gamma),c=c(\gamma),K \ge K(\gamma)$ be parameters of \Cref{lem:many-cycles} so that we can find $cn^\ell$ copies of a colour-alternating cycle of length $4 \le \ell \le L.$ Let $C=C(L,t6^L)\ge C(\ell,t6^L)$ be the parameter given by \Cref{thm:kst}. By assigning each vertex of $V(G)$ into one of $\ell$ parts uniformly at random we can find a partition of $V(G)$ into $V_1,\ldots,V_\ell$ such that there are $cn^\ell/\ell^\ell$ colour-alternating cycles $v_1\ldots v_\ell$ with $v_i \in V_i$.
We also know that at least half of these cycles always use edges of the same colour between all $V_i,V_{i+1}.$ We now build an $\ell$-graph $H$ on the same vertex set as $G$ whose edges correspond to sets of vertices of such colour-alternating cycles. So we know $H$ has at least $\frac{c}{2\ell^\ell}n^\ell  \ge Cn^{\ell-1/(t^\ell\cdot 6^{\ell L})}$ many edges, by taking $K$ large enough, depending on $t,L.$ So \Cref{thm:kst} implies that $H$ contains $K^{(\ell)}(t6^L)$ as a subgraph, which corresponds to a desired $\mathcal{C}(t6^L).$
\end{proof}

\subsection{Ordered graphs}
In our arguments it will not be enough to just find a blow-up of a colour-alternating cycle as in the previous subsection; we will also care about the ``order'' in which the cycles are embedded. In this section we give some notation about ordered graphs and a result which we will need later.

An \emph{ordered graph} is a graph together with a total order of its vertex set.
Here, whenever $G$ is a graph on an indexed vertex set $V(G) = \{v_1, \dots, v_n\}$, we assume that $G$ is ordered by $v_i < v_j \iff i < j$. An \emph{ordered subgraph} of an ordered graph $G$ is a subgraph of $G$ that is endowed with the order that is induced by $G$ and if not stated otherwise, we assume that subgraphs of $G$ are always endowed with that order. For us, two vertices $u < v$ of an ordered graph $G$ are called \emph{neighbouring}, if the set of vertices between $u$ and $v$, that is $\{x \in V(G) | u \leq x \leq v\}$, is either just $\{u,v\}$ or the whole vertex set $V(G)$.

Given an ordered graph $G$ we say a blow-up $H=G(k)$ of $G$ is \textit{ordered consistently} if for any $x,y \in V(H)$ which belong to parts of the blow-up coming from vertices $u,v\in G$ respectively we have $x <_H y$ iff $u <_G v.$

\begin{lem}\label{lem:ordered}
Let $t,L\in \mathbb{N},$ $H$ be a graph on $L$ vertices and $H(t2^L)\subseteq G$ for an ordered graph $G$. There exists an ordering of $H$ for which the consistently ordered $H(t)$ is an ordered subgraph of $G.$
\end{lem}
\begin{proof}
We prove the result by induction on $L,$ where the $L=1$ case is immediate. Let $\{V_1, \dots, V_L\}$ be the clusters of vertices of $H(t2^L),$ so $|V_i|=t2^L.$ Let $w_1, \dots, w_p$ be the median vertices of the sets $V_1, \dots ,V_p$ with respect to the ordering of $H(t2^L)$ induced by $G$ and assume without loss of generality that $w_1$ is the smallest of them. We now throw away all vertices of $V_1$ that are larger than $w_1$ and all vertices of $V_i$ that are smaller than $w_i$ for $i \geq 2$. This leaves us with $L$ sets $\{W_1, \dots, W_L\}$ of size $\ceil{|V_i|/2}=t2^{L-1}$ with the property that $v_1 \in W_1, v_i \in W_i \implies v_1 <_G w_1<_G w_i<_G v_i$ for all $i \geq 2$. If $v\in H$ corresponds to $V_1$ and we denote $H'=H-v$ then $\mathcal{W} = \{W_2, \dots, W_L\}$ spans $H'(t2^{L-1})\subseteq G\setminus V_1$. By the induction hypothesis we can find a consistently ordered $H'(t)$ as an ordered subgraph of $G\setminus V_1$ which together with any subset of size $t$ of $W_1$ gives the desired consistently ordered $H(t)$ in $G$.
\end{proof}

\section{Proof of \Cref{thm:main}}\label{sec:main}

\subsection{Constructing an auxiliary graph}\label{constrA}

Throughout the whole section, let $G$ be a Hamiltonian graph on $n$ vertices. First of all, let us fix a Hamilton cycle $H$ of $G$ and name the vertices of $G$ such that $H = v_1 v_2 \dots v_n v_1$. We assume that $G$ is ordered according to this labelling. Also, let us denote the edges of $H$ by $e_1, e_2, \dots , e_n$ such that $e_1 = v_1 v_2, \dots , e_n = v_n v_1$. In all our following statements, we will identify $v_{n+1}$ and $v_1$, and more generally $v_i$ and $v_j$, as well as $e_i$ and $e_j$, if $i$ and $j$ are congruent modulo $n$. Furthermore, since we can always picture $G$ as a large cycle with some edges inside it, we call all the edges that are not part of $H$, the \emph{inner edges} of $G$.

Our goal is to find a $2$-factor with a fixed number of cycles in $G$. Note that, if $G$ is dense, it is not hard to find a large collection of vertex-disjoint cycles in $G$. The difficulty lies in the fact that we want this collection to be spanning while still controlling the exact number of cycles. Naturally, we have to rely on the Hamiltonian structure of $G$ to give us such a spanning collection of cycles. Indeed, when building these cycles we will try to use large parts of the Hamilton cycle $H$ as a whole and connect them correctly using some inner edges of $G$. It is convenient for our approach to construct an auxiliary graph $A$ out of $G$, that captures the information we need about the inner edges of $G$.

\begin{defn} \label{defn:aux}

Given the setup above, we define the \emph{auxiliary graph} $A = A(G,H)$ as the following ordered, $2$-edge-coloured $n$-vertex graph:

\begin{enumerate}
    \item Every vertex of $A$ corresponds to exactly one edge of $H$, thus we have $V(A) = \{e_1, \dots , e_n\}$ and we order the vertices of $A$ according to this labelling;
    \item two vertices $e_i = v_i v_{i+1}$ and $e_j = v_j v_{j+1}$ of $A$ are connected with a red edge if there is an inner edge of $G$ connecting $v_{i+1}$ and $v_{j+1}$;
    \item similarly, the vertices $e_i$ and $e_j$ of $A$ are connected with a blue edge if there is an inner edge of $G$ connecting $v_i$ and $v_j$.
\end{enumerate}

\end{defn}

 Throughout this section, let $A = A(G,H)$ for our fixed $G$ and $H$. Note that, by the above definition, every edge $\ell \in E(A)$ corresponds to a unique inner edge $e$ of $G$. In the following, we denote this edge by $e(\ell) \in E(G)$. To be precise, if $\ell = e_i e_j$, then $e(\ell) \coloneqq v_{i+1} v_{j+1}$ if $\ell$ is a red edge and $e(\ell) \coloneqq v_i v_j$ if $\ell$ is a blue edge. Conversely, every inner edge of $G$ corresponds to exactly one red edge and to one blue edge of $A$. This leads to the following observation:

\begin{obs} \label{deltaA}
For $i \in \{1, \dots, n\}$, we have $d^{A}_1(e_i) = d^{G}(v_{i+1})-2$ and $d^{A}_2(e_i) = d^{G}(v_i)-2$. In particular, we have $\delta_1(A) = \delta_2(A) = \delta(G) - 2$.
\end{obs}

In \Cref{ExA} we give an example of a Hamiltonian graph and its corresponding auxiliary graph.

\begin{figure}[ht]
    \caption{Let us call the left graph $G$ and fix its Hamilton cycle $H = v_1 \dots v_8 v_1$. Then the graph on the right is the auxiliary graph $A(G,H)$.}
    \includegraphics[scale = 0.8]{2-factors_picture1.pdf}
    \label{ExA}
\end{figure}

The motivation for defining $A$ just as above is given by the fact that 2-regular (possibly non-spanning!) subgraphs $S \subseteq A$ satisfying some extra conditions naturally correspond to a $2$-factor in $G$. Recall that in our setting, two vertices $e_i$ and $e_j$ of $A$ are neighbouring if $|i-j| \equiv 1$ (modulo $n$). Let us make the following definition:

\begin{defn} \label{defn:correspondence}
Given the same setup as above and a subgraph $S \subseteq A$ that is a union of vertex-disjoint colour-alternating cycles without neighbouring vertices (i.e.\ if $e_i \in V(S)$ then $e_{i-1},e_{i+1}\notin V(S)$), we define its corresponding subgraph $F(S) \subseteq G$ as follows:

\begin{enumerate}
    \item $V(F(S)) \coloneqq V(G)$;
    \item the edges of $F(S)$ are all the edges of $H$ except for those that correspond to vertices of $S$. Additionally, for every edge $\ell \in E(S)$, let the corresponding inner edge $e(\ell)$ be an edge of $F(S)$ too. That is, $E(F(S)) \coloneqq \left(\{e_1, \ldots, e_n\} \setminus V(S)\right) \cup \{e(\ell) \mid \ell \in E(S)\}$.
\end{enumerate}
\end{defn}

\begin{lem} \label{correspondence}
If $S \subseteq A$ is a union of vertex-disjoint colour-alternating cycles without neighbouring vertices, then $F(S) \subseteq G$ is a $2$-factor.
\end{lem}

In order to illustrate the above definitions, consider the Hamiltonian graph given in \Cref{ExA} and the subgraphs $S_1$ and $S_2$ of the corresponding auxiliary graph where $S_1$ is just the cycle $e_2 e_4 e_6 e_8 e_2$ and $S_2$ is the union of the cycles $e_1 e_3 e_1$ and $e_5 e_7 e_5$. Their corresponding $2$-factors $F(S_1)$ and $F(S_2)$ are shown as dashed in \Cref{ExCorr}. Note that they use the same inner edges of $G$ but still have different numbers of cycles.

\begin{figure}[ht]
    \caption{$2$-factors $F(S_1)$ and $F(S_2)$ used in the illustration above.}
    \includegraphics[scale = 0.8]{2-factors_picture2.pdf}
    \label{ExCorr}
\end{figure}

\begin{proof}[ of \Cref{correspondence}]
Since $F \coloneqq F(S)$ consists of exactly $n$ edges, it suffices to show that $\delta(F)\ge 2$. Let $v_j$ be an arbitrary vertex of $F$. We distinguish two cases: If both edges $e_{j-1}, e_j \notin V(S)$, then $e_{j-1}, e_j \in E(F)$ and $v_j$ is incident to $e_{j-1}$ and $e_j$ in $F$. Else, exactly one of the edges $e_{j-1}$ and $e_j$ is a vertex of $S$ since $S$ contains no neighbouring vertices. In this case we use the fact that every vertex $e_i$ of $S$ is incident to a red edge $\ell_i$ and to a blue edge $\ell_i'$. Hence, by \Cref{defn:correspondence}, either $e_{j-1}\in S$ and $e_j \notin S$ in which case $v_j$ is incident to $e_j$ and $e(\ell_{j-1})$ in $F$ or $e_{j-1}\notin S$ and $e_j \in S$ in which case $v_j$ is incident to $e_{j-1}$ and $e(\ell_j')$ in $F$. In both cases these two edges are distinct as one of them is an inner edge of $G$ and the other one is not.
\end{proof}

We note that $F(S)$ does not only depend on the structure of $S$ but also on the order in which $S$ is embedded within $A$. However, it is immediate that if $S$ is embedded in auxiliary graphs of two Hamiltonian graphs (possibly with different number of vertices) in the same order then $F(S)$ has the same number of cycles in both cases.

\begin{obs}\label{obs:samecomps}
Let $A_1=A(G_1,H_1)$ and $A_2=A(G_2,H_2)$. Let $S_1$ and $S_2$ be disjoint unions of colour-alternating cycles without neighbouring vertices, which are isomorphic as coloured subgraphs of $A_1$ and $A_2$ whose corresponding vertices appear in the same order along $H_1$ and $H_2.$ Then $F(S_1)$ and $F(S_2)$ consist of the same number of cycles.
\end{obs}

We remark that it is not always true that all $2$-factors of $G$ arise as $F(S)$ for some $S \subseteq A.$

\subsection{Controlling the number of cycles} \label{control}

It is not hard to see that the auxiliary graph $A$ (of a graph with a big enough minimum degree) must contain a colour-alternating cycle $C$, which corresponds to a $2$-factor $F(C) \subseteq G$ by \Cref{correspondence} (disregarding, for the moment, the issue of $C$ containing neighbouring vertices). However, it is not at all obvious how to generally determine the number of components of $F(C).$
We begin by giving a rough upper bound.

\begin{obs} \label{obs:no.cycles}
If $C \subseteq A$ is a non-empty colour-alternating cycle of length $L$ without neighbouring vertices, then the number of components of the corresponding $2$-factor $F(C)$ is at most $L$.
\end{obs}

\begin{proof}
Note that the $2$-factor $F(C)$ contains exactly $L$ inner edges and, since $F(C) \neq H$, each cycle of $F(C)$ must contain at least one inner edge (in fact, at least two in our setting).
\end{proof}

However, in order to prove \Cref{thm:main}, we need to be able to show the existence of a $2$-factor consisting of exactly $k$ cycles, for a fixed predetermined number $k$. This is where we are going to make use of \Cref{lem:cycle-blow-up,lem:ordered} which allow us to find a consistently ordered blow-up of $C$.  This will give us the freedom to find slight modifications of $C$ with different numbers of cycles in $F(C)$.

\subsubsection{Going up}
In this subsection we give a modification of a union of colour-alternating cycles which will have precisely one more cycle in its corresponding $2$-factor.
\begin{defn}\label{defn:goingup}
Let $S$ be a disjoint union of colour-alternating cycles with $V(S)=\{s_1, \dots, s_m\}$ and let $C$ be a cycle of $S$. We construct a $2$-edge-coloured ordered graph $U(S,C)$ as follows:
\begin{enumerate}
    \item Start with a copy of $S$ and for every $s_i \in V(C)$, add a vertex $s_{i+1/2}$;
    \item For every red or blue edge $s_i s_j \in E(C)$, add an edge $s_{i+1/2} s_{j+1/2}$ of the same colour;
    \item Order the resulting graph according to the order of the indices of its vertices.
\end{enumerate}
Given a $2$-edge-coloured ordered graph $U$, we say that $U$ is a \emph{going-up version} of $S$, if there exists a component $C$ of $S$ such that $U$ and $U(S,C)$ are isomorphic $2$-edge-coloured ordered graphs.
\end{defn}

In other words $U(S,C)$ consists of $S$ with an additional copy of $C$ ordered in such a way that the vertices of the new copy of $C$ immediately follow their corresponding vertices of the original copy of $C$. In particular, $U$ is also a disjoint union of colour-alternating cycles and is an ordered subgraph of a consistently ordered $S(2)$. Note if $S$ contains no double edges, neither does $U.$

\Cref{fig:goingup} shows what a going-up version $U$ of $S$ looks like if $S$ is just a colour-alternating $C_4$. \Cref{fig:goingupG} shows what the corresponding $2$-factors look like (assuming $S \subseteq U \subseteq A$). Note that the dashed cycles of $F(U)$ have the same structure as the dashed cycles in $F(S)$ but $F(U)$ additionally has a new bold cycle. We now show that a similar situation occurs in general.

\begin{figure}[ht]
    \caption{A colour-alternating cycle $S$ and a going-up version of it $U$}
    \includegraphics[scale = 0.8]{2-factors_going_up.pdf}
    \label{fig:goingup}
\end{figure}

\begin{figure}[ht]
    \caption{$2$-factors corresponding to $U$ and $S$ given in \Cref{fig:goingup}.}
    \includegraphics[scale = 0.8]{2-factors_going_up_G.pdf}
    \label{fig:goingupG}
\end{figure}

\begin{lem}[Going up]\label{lem:goingup}
Let $S \subseteq A$ be a disjoint union of colour-alternating cycles without neighbouring vertices and let $U$ be an ordered subgraph of $A$ without neighbouring vertices that is a going-up version of $S$. Then, the $2$-factor $F(U) \subseteq G$ has exactly one component more than $F(S)$.
\end{lem}

\begin{proof}[ of \Cref{lem:goingup}]
For an edge $e=v_k v_{k+1}\in H$ we let $v^{+}(e)=v_{k+1}$ and $v^{-}(e)=v_k.$
We denote the vertices of $S$ by $s_1, \dots, s_m$ according to their order in $A$. Let $C$ be a colour-alternating cycle $s_{j_1} \dots s_{j_k} s_{j_1}$ in $S$ for which $U=U(S,C)$. Let us denote the vertices of $U$ by $u_1, \dots, u_m$ and $u_{j_1 + 1/2}, \dots, u_{j_k+1/2}$ as they appear along $H$ such that $u_1, \dots, u_m$ make a copy of $S$ and $u_{j_1}, \dots, u_{j_k}$ correspond to $C$.
The vertices $v^+(u_{j_i})$ and $v^-(u_{j_i + 1/2})$ are connected in $F(U)$ by paths $P_i \subseteq H$ for $i \in \{1, \dots, k\}$. Furthermore, since $C$ is a colour-alternating cycle either $v^+(u_{j_i})v^+(u_{j_{i+1}})\in E(G)$ for all odd $i$ and $v^-(u_{j_i + 1/2})v^-(u_{j_{i+1} + 1/2})\in E(G)$ for all even $i$ or vice versa in terms of parity. This means that taking all $P_i$ and these edges we obtain one cycle $$Z:=v^+(u_{j_1})v^+(u_{j_2})P_2v^-(u_{j_2+1/2})v^-(u_{j_3+1/2})P_3v^{+}(u_{j_3})\dots P_kv^-(u_{j_k + 1/2})v^-(u_{j_1+1/2})P_1v^+(u_{j_1})\in F(U),$$ if $C$ starts with a red edge (which is exactly the bold cycle in the example shown in \Cref{fig:goingupG}) or
$$Z:=v^-(u_{j_1+1/2})v^-(u_{j_2+1/2})P_2v^+(u_{j_2})v^+(u_{j_3})P_3v^{-}(u_{j_3+1/2})\dots P_kv^+(u_{j_k})v^+(u_{j_1})P_1v^-(u_{j_1+1/2})\in F(U),$$
if $C$ starts with a blue edge.

Let us now consider the graph $G'$ that is obtained from $G$ by deleting $Z$ (including all edges incident to vertices of $Z$) and adding the edges $S_{j_i} = v^-(u_{j_i}) v^+(u_{j_i + 1/2})$ for $i \in \{1, \dots, k\}$. Let $H'$ be the Hamilton cycle of $G'$ made of $H$ and $S_j$'s ordered according to the order of $G$. We claim that sending the vertices $s_i$ to $S_i$ if $s_i \in C$ and to $u_i$ otherwise for $i \in \{1, \dots, m\}$ gives an order-preserving isomorphism from $S$ to its image $S' \subseteq A(G', H')$. Indeed, if $s_i, s_j \notin C$, then the fact that $u_i u_j$ is a red or a blue edge whenever $s_i s_j$ is a red or a blue edge just follows from \Cref{defn:goingup}. Furthermore, if $s_{j_i} s_{j_{i+1}}$ is a red edge for $i \in \{1, \dots, k\}$, then $v^+(s_{j_i + 1/2})$ is adjacent to $v^+(s_{j_{i+1}+1/2})$, which means that $S_i S_{i+1}$ is a red edge. This works analogously for blue edges of $C$, which shows the claim. Hence, by \Cref{obs:samecomps}, the $2$-factor $F(S')$ in $G'$ has the same number of components as $F(S)$ in $G$. However, since $F(S')$ is by definition just $F(U) \setminus Z$, this completes the proof.
\end{proof}

\subsubsection{Going down}
We now turn to the remaining case when we want to find a $2$-factor with less components than one that we already found.

\begin{defn}
Let $S \subseteq A$ be a disjoint union of colour-alternating cycles without neighbouring vertices. We say that a vertex $e_k \in V(A)$ \emph{separates components of $F(S)$} if the vertices $v_k$ and $v_{k+1}$ lie in different connected components of $F(S)$.
\end{defn}

\begin{obs}\label{obs:sepcomps}
If $F(S)$ has more than one connected component, then at least one vertex of $S$ separates components.
\end{obs}
\begin{proof}
Since $F(S)$ is not connected there must exist vertices $v_k,v_{k+1}$ of $H$ belonging to different components of $F(S).$ Let $e_k=v_k v_{k+1}$ so $e_k \notin E(F(S))$. Since the only edges of $H$ (that is vertices of $A$) that are not in $E(F(S))$ are vertices of $S,$ $e_k$ is the desired separating vertex.
\end{proof}

We are now ready to construct a going-down version of $S$ giving rise to a $2$-factor with one less cycle.

\begin{defn}\label{defn:goingdown}
Let $S$ be a disjoint union of colour-alternating cycles with $V(S)=\{s_1, \dots, s_m\}$. For any $s_k \in V(S)$ we construct the $2$-edge-coloured ordered graph $D=D(S,s_k)$ as follows:
\begin{enumerate}
    \item Start with a copy of $S$ and for every vertex $s_i$ in the cycle $C\subseteq S$ that contains $s_k$, add the vertices $s_{i+1/3}$ and $s_{i+2/3}$ to $D$;
    \item if $i, j \neq k$ and if $s_i s_j$ is a red or a blue edge of $S$, then add the edges $s_{i+1/3}s_{j+1/3}$ and $s_{i+2/3}s_{j+2/3}$ of the same colour to $D$;
    \item if $s_i s_k$ is the blue edge of $S$ incident to $s_k$, then delete it and add the blue edges $s_i s_{k+1/3}$, $s_{i+1/3} s_{k+2/3}$ and $s_{i+2/3} s_k$ to $D$;
    \item if $s_i s_k$ is the red edge of $S$ incident to $s_k$, then add the red edges $s_{i+1/3} s_{k+2/3}$ and $s_{i+2/3} s_{k+1/3}$ to $D$;
    \item order the resulting graph according to the order of the indices of its vertices.
\end{enumerate}
Let $S\subseteq A$ be a disjoint union of colour alternating cycles without neighbouring vertices, so that $F(S)$ exists. We say that a $2$-edge-coloured ordered graph $D$ is a \emph{going-down version} of $S$ if there exists a vertex $s_k$ that separates components of $F(S)$ such that $D$ and $D(S,s_k)$ are isomorphic $2$-edge-coloured ordered graphs.
\end{defn}

In other words $D=D(S,s_k)$ consists of a copy of $S$ with added two copies of the cycle containing $s_k$ where the edges incident to $s_k$ and its copies are rewired in a certain way. It is easy to see that every vertex of $D$ is still incident to exactly one edge of each colour so is still a disjoint union of colour-alternating cycles. Note also that $D$ is an ordered subgraph of consistently ordered $S(3).$ If $S$ contained no double edges neither does $D$.

\Cref{fig:goingdown} shows a going-down version $D = D(S, s_1)$ for $S$ on $\{s_1, \dots, s_4\}$ being again a colour-alternating $C_4$. Note that $F(D)$, shown in \Cref{fig:goingdownG}, contains two paths, marked as dotted and bold, that connect the two dashed parts of $F(D)$ that resemble the two disjoint cycles of $F(S),$ into a single cycle. We will show that this occurs in general.

\begin{figure}[ht]
    \caption{A colour-alternating cycle $S$ and a going-down version of it $D(S,s_1).$}
    \includegraphics[scale = 0.8]{2-factors_going_down.pdf}
    \label{fig:goingdown}
\end{figure}

\begin{figure}[ht]
    \caption{$2$-factors corresponding to $U$ and $D(S,s_1)$ given in \Cref{fig:goingdown}.}
    \includegraphics[scale = 0.8]{2-factors_going_down_G.pdf}
    \label{fig:goingdownG}
\end{figure}

\begin{lem}[Going down]\label{lem:goingdown}
Let $S \subseteq A$ be a disjoint union of colour-alternating cycles without neighbouring vertices and let $D$ be an ordered subgraph of $A$ without neighbouring vertices that is a going-down version of $S$. Then the $2$-factor $F(D) \subseteq G$ consists of one cycle less than $F(S)$.
\end{lem}

\begin{proof}
For an edge $e=v_kv_{k+1}\in H$ we let $v^{+}(e)=v_{k+1}$ and $v^{-}(e)=v_k.$
We denote the vertices of $S$ by $s_1, \dots, s_m$   where $D=D(S,s_1)$ and $s_1$ separates components of $F(S)$. We denote the vertices of $D$ by $d_1, \dots, d_m$ and $d_{4/3}, d_{5/3}, d_{j_1+1/3}, d_{j_1 + 2/3}, \dots, d_{j_k + 2/3}$ as they appear along $H$ such that $d_1, \dots, d_m$ make a copy of $S$ in which $d_1$ corresponds to $s_1$ and $d_1,d_{j_1}, \dots, d_{j_k}$ to the cycle $C = s_1 s_{j_1} \dots s_{j_k} s_1$ of $S.$

The vertices $v^+(d_{j_i})$ and $v^-(d_{j_i + 1/3})$ as well as the vertices $v^+(d_{j_i+1/3})$ and $v^-(d_{j_i+2/3})$ in $F(D)$ are connected by paths $P_i \subseteq H$ and $Q_i \subseteq H$ respectively for all $i \in \{1, \dots, k\}$.

If $C$ begins by a red edge then $$P:=v^+(d_1)v^+(d_{j_1})P_1v^-(d_{j_1+1/3})v^-(d_{j_2+1/3})P_2v^{+}(d_{j_2})\dots P_kv^-(d_{j_k + 1/3})v^-(d_{5/3})\in F(D),$$ where $v^+(d_1)v^+(d_{j_1})\in F(D)$ by \Cref{defn:goingdown} part 4; $v^-(d_{j_k + 1/3})v^-(d_{5/3})\in F(D)$ by part $3$ and edges between paths $P_i$ are in $F(D)$ by part 2 in the same way as in the going up case. Similarly, $$Q:=v^+(d_{5/3})v^+(d_{j_1+1/3})Q_1v^-(d_{j_1+2/3})v^-(d_{j_2+2/3})Q_2v^{+}(d_{j_2+1/3})\dots Q_kv^-(d_{j_k +2/3})v^-(d_{1})\in F(D)$$
On the other hand if $C$ begins by a blue edge then we have
$$P:=v^-(d_{5/3})v^-(d_{j_1+1/3})P_1v^+(d_{j_1})v^+(d_{j_2})P_2\dots P_kv^+(d_{j_k})v^+(d_{1})\in F(D),$$
$$Q:=v^-(d_{1})v^-(d_{j_1+2/3})Q_1v^+(d_{j_1+1/3})v^+(d_{j_2+1/3})Q_2\dots Q_kv^+(d_{j_k+1/3})v^+(d_{5/3})\in F(D)$$
So in either case the path $P\subseteq F(D)$ contains $P_1, \dots, P_k$ and has endpoints $v^+(d_1),v^-(d_{5/3})$ while $Q \subseteq F(D)$ contains $Q_1, \dots, Q_k$ and has endpoints $v^+(d_{5/3}),v^-(d_1)$. For example in \Cref{fig:goingdownG}, the paths $P$ and $Q$ correspond to the dotted and the bold path respectively.

Our goal now is to show that $P$ and $Q$ connect two ``originally distinct'' components that are ``inherited'' from $F(S)$. Consider the graph $G'$ that is obtained from $G$ by deleting all the vertices of paths $P_i$ and $Q_i$ (equivalently all inner vertices of $P$ and $Q$) and adding the edges $S_{j_i} = v^-(d_{j_i}) v^+(d_{j_i + 2/3})$ for $i \in \{1, \dots, k\}$. Let $H'$ be the Hamilton cycle of $G'$ made of $H$ and $S_j$'s ordered according to the order of $G$.  First, we claim that the map that sends $s_1$ to $d_{4/3}$ and $s_i$ to $S_i$ if $s_i$ is part of $C \setminus \{s_1\}$ and to $d_i$ otherwise for $i \in \{2, \dots, m\}$ is an order-preserving isomorphism from $S$ onto its image $S' \subseteq A(G', H')$. Indeed, by \Cref{defn:goingdown} parts 3 and 4 for $i=1,k$ if $s_1s_{j_i}$ is red then $d_{4/3}d_{j_i+2/3}$ is a red edge of $A$ so $v^+(d_{4/3})v^+(d_{j_i+2/3})\in F(D)$ implying that $d_{4/3} S_{j_i}$ is red in $A'.$ If $s_1s_{j_i}$ is blue then $d_{4/3}d_{j_i}$ is a blue edge of $A$ so $v^-(d_{4/3})v^-(d_{j_i})\in F(D)$ implying that $d_{4/3} S_{j_i}$ is blue in $A'.$
For $i\neq k$ edge $S_{j_i} S_{j_{i+1}}$ is of the same colour as $s_{j_i}s_{j_{i+1}}$ by \Cref{defn:goingdown} part 2 and for $s_i,s_j \notin  C$ we know $d_i d_j$ has the same colour by part 1. Therefore, by \Cref{obs:samecomps}, $F(S')$ has the same number of components as $F(S)$. Since $s_1$ separates components in $S$ we know that $d_{4/3}$ separates components in $F(S')$. This means in particular that $d_1$ and $d_{5/3}$ lie in two different cycles $C_1$ and $C_2$ of $F(S')$. Now, observe that we obtain $F(D)$ from $F(S')$ by deleting $d_1$ and $d_{5/3}$ and adding the paths $P$ and $Q$. However, since $P$ connects $v^+(d_1)$ and $v^-(d_{5/3})$ and $Q$ connects $v^+(d_{5/3})$ and $v^-(d_1)$, this process joins $C_1$ and $C_2$ into one big cycle and hence, $F(D)$ has exactly one component less than $F(S)$.
\end{proof}

\subsection{Completing the proof}\label{subs:complete}

We are now ready to put all the ingredients together in order to complete our proof of \Cref{thm:main} in the way that has already been outlined throughout the previous section.

\begin{proof}[ of \Cref{thm:main}]
Let $k$ be a positive integer and $\varepsilon$ a positive real number. Let $L=L(\varepsilon/2),K=K(\varepsilon/2,2^k)$ be the parameters coming from \Cref{lem:cycle-blow-up}. Let $N\ge \max(4/\varepsilon,K).$

Now, suppose that $G$ is a Hamiltonian graph on $n \geq N$ vertices with minimum degree $\delta(G) \geq \varepsilon n$. Let us fix a Hamilton cycle $H \subseteq G$, name the vertices of $G$ such that $H = v_1 v_2 \dots v_n v_1$ and assume that $G$ is ordered according to this labelling. Let $A = A(G,H)$ be the ordered, $2$-edge-coloured auxiliary graph corresponding to $G$ and $H$ according to \Cref{defn:aux}. We know by \Cref{deltaA} that $\delta_\nu(A)\ge \delta_\nu(G)-2 \ge \frac{\varepsilon}{2} n.$

\Cref{lem:cycle-blow-up} shows that there is a $\mathcal{C}(2^k6^L)\subseteq A$ where $\mathcal{C}$ is a colour-alternating cycle of length at most $L$ without double-edges. \Cref{lem:ordered} allows us to find a consistently ordered $\mathcal{C}(2^k3^L)$ as an ordered subgraph of $A.$ By removing every second vertex of $\mathcal{C}(2^k3^L)$ in $A$ we obtain a consistently ordered $\mathcal{C}'=\mathcal{C}(2^{k-1}3^L)$ that is an ordered subgraph of $A$ without neighbouring vertices. For $\mathcal{C}\subseteq \mathcal{C}'$  by \Cref{correspondence} we obtain a $2$-factor $F(\mathcal{C}) \subseteq G$. Let $\ell$ be the number of cycles of $F(\mathcal{C}).$ By \Cref{obs:no.cycles}, we know that $1 \leq \ell \leq L$.

Let us first assume that $k > \ell$. We find a sequence $S_0, S_1, \dots, S_{k-\ell}$ defined as follows: let $S_0 = \mathcal{C};$ given $S_{i-1}$ let $\mathcal{C}_{i-1}$ be an arbitrary cycle of $S_{i-1}$ and let $S_{i} = U(S_{i-1}, \mathcal{C}_{i-1})$. By construction, $S_i$ is again a disjoint union of colour-alternating cycles, without double edges, and is an ordered subgraph of $\mathcal{C}(2^i) \subseteq \mathcal{C}'$ (since by construction $S_i\subseteq S_{i-1}(2)$). Therefore, for all $i\le k-\ell$ there is an order-preserving embedding of $S_i$ into $A$ without neighbouring vertices. So, by \Cref{correspondence} and \Cref{lem:goingup} we deduce that $F(S_i)$ has one more cycle than $F(S_{i-1})$. In particular, the $2$-factor $F(S_{k-\ell}) \subseteq G$ consists of exactly $k$ components.

Let us now assume that $k < \ell$. Here, we find a sequence $S_0, S_1, \dots, S_{\ell - k}$ of disjoint unions of colour-alternating cycles that are ordered subgraphs of $A$ without neighbouring vertices such that $F(S_i)$ consists of $\ell-i$ cycles. Let $S_0 = \mathcal{C},$ and assume we are given $S_{i-1}$ for $i\le \ell-k$ with $F(S_{i-1})$ having $\ell-i+1 \ge k+1 \ge 2$ cycles. This means that $S_{i-1}$ has a vertex $v_{i-1}$ that separates components of $F(S_{i-1})$ by \Cref{obs:sepcomps}. We let $S_i = D(S_{i-1}, v_{i-1})$, which is a disjoint union of colour-alternating cycles, without double edges, and is an ordered subgraph of a consistently ordered $\mathcal{C}(3^i)$ (since by construction $S_i \subseteq S_{i-1}(3)$). Note that $\ell - k \leq L$ by \Cref{obs:no.cycles} and hence, $ \mathcal{C}(3^i) \subseteq \mathcal{C}(3^{\ell - k}) \subseteq \mathcal{C}'$ so we can find a copy of $S_i$ into $A$ without having neighbouring vertices. By \Cref{lem:goingdown}, $F(S_i)$ has one less cycle than $F(S_{i-1})$, so exactly $\ell - i$ cycles. In particular, $F(S_{\ell - k})$ is a $2$-factor in $G$ with $k$ cycles, which concludes the proof.
\end{proof}

\section{Concluding remarks and open problems}
In this paper we show that in a Hamiltonian graph the minimum degree condition needed to guarantee any $2$-factor with $k$-cycles is sublinear in the number of vertices. The best lower bound is still only a constant.
In the case of a $2$-factor with two components, the best bounds are given by Faudree et al. \cite{Faudree05} who construct minimum degree $4$ Hamiltonian graphs without a $2$-factor with $2$ components. In the case of $2$-factors with $k$ components, no constructions have been given previously, but it is easy to see that a minimum degree of at least $k+2$ is necessary:
\begin{prop}
There are arbitrarily large Hamiltonian graphs with minimum degree $k+1$ which do not have a $2$-factor with $k$ components.
\end{prop}
\begin{proof}
Let $G$ consist of a cycle $\mathcal{C}$ of length $n-k+1$ and an independent set $U$ of size $k-1$ with all the edges between $\mathcal{C}$ and $U$ added. It's easy to see that for $n\ge 2k$,  $G$ is Hamiltonian and has minimum degree $k+1$. However $G$ does not have a $2$-factor with $k$ components (e.g.\ because every cycle in a $2$-factor of $G$ must use at least one vertex in $U$).
\end{proof}
For fixed $k$, we do not know of any Hamiltonian graphs with non-constant minimum degree which do not have a $2$-factor with $k$ components. This indicates that the necessary minimum degree in \Cref{conj:main} may in fact be much smaller, perhaps even a constant (depending on $k$). A step in this direction was made by Pfender \cite{Pfender04} who showed that in the $k=2$ case, a Hamiltonian graph $G$ with minimum degree of $7$ contains a $2$-factor with $2$ cycles in a very special case when $G$ is claw-free.

If one takes greater care with various parameters in \Cref{sec:prelim} one can show that a minimum degree of $\frac{Cn}{\sqrt[4]{\log \log n / (\log \log \log n)^2}}$ suffices for finding an ordered blow-up of a short cycle so in particular this minimum degree is enough to find $2$-factors consisting of a fixed number of cycles. We believe that it would be messy but not too hard to improve this a little bit further, but to reduce the minimum degree condition to $n^{1-\varepsilon}$ would require some new ideas. On the other hand we do believe that our approach of finding alternating cycles in the auxiliary graph could still be useful in this case, but one needs to either find a better way of finding ordered blow-ups of short cycles or obtain a better understanding of how the number of cycles in $F(S)$ depends on the order and structure of a disjoint union of colour-alternating cycles $S.$ Another possibility is to augment the auxiliary graph in order to include edges that connect the front/back to the back/front vertex of two edges of the Hamilton cycle, which would allow us to obtain a $1$-to-$1$-correspondence between $2$-factors of $G$ and suitable structures in this new auxiliary graph.

Another way of saying that a graph is Hamiltonian is that it has a $2$-factor consisting of a single cycle. A possibly interesting further question which arises is whether knowing that $G$ contains a $2$-factor consisting of $\ell$ cycles already allows the minimum degree condition needed for having a $2$-factor with $k>\ell$ cycles to be weakened.

\textbf{Acknowledgements.} We are extremely grateful to the anonymous referees for their careful reading of the paper and many useful suggestions.