Hamilton cycles in highly connected and expanding graphs

In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on two properties only: one requiring expansion of "small” sets, the other ensuring the existence of an edge between any two disjoint "large” sets. We also discuss applications in positional games, random graphs and extremal graph theor

Logconcave Random Graphs

We propose the following model of a random graph on n vertices. Let F be a distribution in Rn(n−1)/2+ with a coordinate for every pair ij with 1≤i,j≤n. Then GF,p is the distribution on graphs with n vertices obtained by picking a random point X from F and defining a graph on nvertices whose edges are pairs ij for which Xij≤p. The standard Erdős-Rényi model is the special case when F is uniform on the 0-1 unit cube. We examine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the Xij are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight.</p