13358572
Avoidance of a giant component in half the edge set of a random graph
Let e1, e2, . . . be a sequence of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e. we sample with replacement). Our goal is to choose, for m as large as possible, a subset E ⊆ {e1, e2, . . . , e2m}, |E| = m, such that the size of the largest component in G = ([n], E) is o(n) (i.e. G does not contain a giant component). Furthermore, the selection process must take place on-line; that is, we must choose to accept or reject an ei based on the previously seen edges e1, . . . , ei−1. We describe an on-line algorithm that succeeds whp for m = .9668n. Furthermore, we find a tight threshold for the off-line version of this question; that is, we find the threshold for the existence of m out of 2m random edges without a giant component. This threshold is m = cn where c satisfies a certain transcendental equation, c ∈ [.9792, .9793]. We also establish new upper bounds for more restricted Achlioptas processes.
19308721
Maximum matchings in a class of random graphs
Let Vn = {1,2,…,n} and D(n, m) be the set of digraphs with vertex set Vn in which each v ϵ Vn has outdegree m. D(n, m) is chosen uniformly at random from D(n, m) and then D(n, m) is obtained by ignoring the orientation of the edges of D(n, m). We show that 
Limn → ∞n evenPr(D(n, 1) has a perfect matching) = 0 
, 
Limn → ∞n evenPr(D(n, 2) has a perfect matching) = 1 
.
6378032
Adding random edges to dense graphs
This paper investigates the addition of random edges to arbitrary dense graphs; in particular, we determine the number of random edges required to ensure various monotone properties including the appearance of a fixed size clique, small diameter and k-connectivity.
36434433
Hamiltonian cycles in random regular graphs
The study of the properties of random graphs was initiated by Erdos and RCnyi [2]. There they considered the asymptotic properties of random graphs with n labelled vertices and M(n) edges as n -+ 00. One problem they left open was the number of edges needed to ensure the almost certain existence of a Hamiltonian cycle. This was later partially solved by Posa [8] and then completely solved by Korshunov [6] and Komlos and Szemeridi [5]. The corresponding result for digraphs follows from a theorem of McDiarmid [7]. An interesting feature of this result, highlighted in [5], is that Hamiltonian cycles “appear” when M(n) is large enough to “ensure” that the graph has no vertices of degree one or zero. It is therefore of interest to study classes of graphs in which the minimal vertex degree exceeds some prescribed value. In [4] we studied random graphs obtained by taking a random digraph in which each vertex has outdegree m and then ignoring the orientation of the arcs. We showed that there exists mO m, is kept fixed and n + a, almost all such graphs have Hamiltonian cycles. We conjecture that the minimum value for m, is 3. In this paper we consider random vertex-labelled regular graphs of degree r and prove a similar result, viz that there exists r,, Q 796 such that, if r > rO 103
6616465
Multi-Coloured Hamilton Cycles In Random Edge-Coloured Graphs
We define a space of random edge-coloured graphs Gn,m,κ which correspond naturally to edge κ-colourings of Gn,m. We show that there exist constants K0, K1 ≤ 21 such that, provided m ≥ K0n log n and κ ≥ K1n, then a random edge-coloured graph contains a multi-coloured Hamilton cycle with probability tending to 1 as the number of vertices n tends to infinity.
27799444
Edge-colouring random graphs
Let pn denote the proportion of n-vertex graphs in class 2, that is, such that the chromatic index exceeds the maximum vertex degree. We extend a result of Erdős and Wilson, and show that n−(12 + e)n < pn < n−(18 − e)n for n sufficiently large.
12252849
On the connectivity of randomm-orientable graphs and digraphs
We consider graphs and digraphs obtained by randomly generating a prescribed number of arcs incident at each vertex.We analyse their almost certain connectivity and apply these results to the expected value of random minimum length spanning trees and arborescences.We also examine the relationship between our results and certain results of Erdős and Rényi.