Asymptotic Behaviour of the Number of Eulerian Circuits

Let G be a simple connected graph all of whose vertices have even degree. A Eulerian circuit in G is a closed walk (see, for example, [2]) which uses every edge of G exactly once. We let Eul(G) denote the number of these up to cyclic equivalence. Our purpose in this paper is to estimate Eul(G) for those G having large algebraic connectivity. Our method is to adopt the proof given in [6] for the case G = Kn. We refer to that paper for the interesting history of this problem, and suggest that readers who want to understand our proofs carefully may find it helpful to have a copy at hand. Since the publication of [6], the work [3] has appeared showing that counting the number of Eulerian circuits in an undirected graph is complete for the class #P . Thus this problem is difficult in terms of complexity theory. Here is an outline of the paper. The asymptotic formula for Eul(Kn) and our main result are presented and discussed in Section 2. In Section 3 we prove some basic properties of the Laplacian matrix, which may be of independent interest. In Section 4 we express Eul(G) in terms of an n-dimensional integral using Cauchy’s formula. The value of the integral is estimated in Sections 5 and 6, using some Lemmas proved in Section 8. We prove the main result in Section 7.