Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Short cycles in random regular graphs. | Short cycles in random regular graphs Brendan D. McKay Department of Computer Science Australian National University Canberra ACT 0200 Australia bdm@ Nicholas C. Wormald and Beata Wysocka Department of Mathematics and Statistics University of Melbourne Vic 3010 Australia nwormald@ beata@ Submitted Aug 10 2003 Accepted May 20 2004 Published Sep 20 2004 Mathematics Subject Classifications 05C80 05C38 05C30 Abstract 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. 1 Introduction Let H be a fixed graph. The asymptotic distribution of the number of subgraphs of a random graph isomorphic to H has been studied in various places such as by Rucinski 9 for the random graph model G n p and Janson 4 for the model G n m . In this paper we consider the distribution in a random d-regular graph. Here and henceforth in the paper random refers to the uniform distribution on the set of all labelled graphs in the specified class. Research supported by the Australian Research Council 1 Research supported by the Australian Research Council. Current address Department of Combinatorics and Optimization University of Waterloo Waterloo ON Canada N2L 3G1. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R66 1 Many properties of random d-regular graphs on n vertices are known see Bollobas 2 or Wormald 11 for details . For d fixed or growing slowly as a function of n such a graph looks like a tree in the neighbourhood of almost all vertices the expected number of cycles of any fixed length is small and for any fixed graph H with more edges than vertices the expected .
