13636326
An Algorithm for Finding Hamilton Cycles in Random Directed Graphs
Abstract We describe a polynomial ( O ( n 1.5 )) time algorithm DHAM for finding hamilton cycles in digraphs. For digraphs chosen uniformly at random from the set of digraphs with vertex set {1, 2, …, n } and m = m ( n ) edges the limiting probability (as n → ∞) that DHAM finds a hamilton cycle equals the limiting probability that the digraph is hamiltonian. Some applications to random “travelling salesman problems” are discussed.
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.
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.
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.
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
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.