Properties of Random Difference Graphs

Abstract

Generate a bipartite graph on a partitioned set of vertices by randomly assigning to each vertex $v$ some weight $w(v) \in [0,1]$ and adding an edge between vertices $u$ and $v$ (in distinct parts) if and only if $w(v) + w(v) > 1$; the results of such processes are known as difference graphs.

Random difference graphs of a given size can be produced either by uniformly random generation of weights or by choosing a graph uniformly at random from the set of all such graphs. We prove that these two methods give rise to the same distribution, and use this equivalence to find exact results for the likelihood of connectivity and Hamiltonicity. We also find the distribution of other properties, such as matching number and degeneracy.