5280068
Near-perfect Token Distribution
Suppose that n tokens are arbitrarily placed on the n nodes of a graph. At each parallel step one token may be moved from each node to an adjacent node. An algorithm for the near-perfect token distribution problem redistributes the tokens in a minimum number of steps, so that, at the end, no more than O(1) tokens reside at each node. (In perfect distribution, at the end, exactly one token resides at each node.)
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.
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 
.
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.
12475992
On Markov Chains for Randomly H-Coloring a Graph
Let H=(W,F) be a graph without multiple edges, but with the possibility of having loops. Let G=(V,E) be a simple graph. A homomorphism c is a map c:V?W with the property that (v,w)?E implies that (c(v),c(w))?F. We will often refer to c(v) as the color of v and c as an H-coloring of G. We consider the problem of choosing a random H-coloring of G by Markov chain Monte Carlo. The probabilistic model we consider includes random proper colorings, random independent sets, and the Widom?Rowlinson and Beach models of statistical physics. We prove negative results for uniform sampling and a positive result for weighted sampling when H is a tree.
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.