Cycle lengths in sparse random graphs

Abstract

AbstractWe study the set of lengths of all cycles that appear in a random d‐regular graph G on n vertices for fixed, as well as in binomial random graphs on n vertices with a fixed average degree . Fundamental results on the distribution of cycle counts in these models were established in the 1980s and early 1990s, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in n. Here we derive, for a random d‐regular graph, the limiting probability that simultaneously contains the entire range for , as an explicit expression which goes to 1 as . For the random graph with , where for some absolute constant , we show the analogous result for the range , where is the length of a longest cycle in G. The limiting probability for coincides with from the d‐regular case when c is the integer . In addition, for the directed random graph we show results analogous to those on , and for both models we find an interval of consecutive cycle lengths in the slightly supercritical regime .