Short Cycles in Random Regular Graphs

Consider random regular graphs of order n and degree d = d(n) 3. Let g = g(n) 3 satisfy (d-1)2g-1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs.

• Type: Journal
• Year: 2004
• Where: COMBINATORICS
• Authors: Brendan D. McKay, Nicholas C. Wormald, Beata Wysocka
• Keywords: COMBINATORICS 2004, Independent Poisson Variables, Random Regular Graphs, Regular