Asymptotic Enumeration of Eulerian Circuits in the Complete Graph

Authors:
Brendan D. McKay;Robert W. Robinson
Affiliations:
Department of Computer Science, Australian National University, Canberra, ACT 0200, Australia (e-mail: bdm@cs.anu.edu.au);Department of Computer Science, University of Georgia, Athens, GA 30602-7404, USA (e-mail: rwr@cs.uga.edu)
Venue:
Combinatorics, Probability and Computing
Year:
1998

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Abstract

We determine the asymptotic behaviour of the number of Eulerian circuits in a complete graph of odd order. One corollary of our result is the following. If a maximum random walk, constrained to use each edge at most once, is taken on Kn, then the probability that all the edges are eventually used is asymptotic to e3/4n−½. Some similar results are obtained about Eulerian circuits and spanning trees in random regular tournaments. We also give exact values for up to 21 nodes.