Asymptotic enumeration of Eulerian circuits for graphs with strong mixing properties

Let G be a simple connected graph all of whose vertices have even degrees. An Eulerian circuit in G is a closed walk (see, for example, [2]) which uses every edge of G exactly once. Two Eulerian circuits are called equivalent if one is a cyclic permutation of the other. It is clear that the size of such an equivalence class equals the number of edges of graph G. Let EC(G) denote the number of equivalence classes of Eulerian circuits in G. The problem of counting the number of Eulerian circuits in an undirected simple graph (i.e. graph without loops and multiple edges) is complete for the class #P , see [3]. Thus this problem is difficult in terms of the complexity theory. Moreover, it should be noted that in contrast to many other hard problems of counting on graphs (see, for example, [1], [9]), even approximate and probabilistic polynomial algorithms for counting the number of Eulerian circuits are not known in the literature. As concerns the class of complete graphs Kn, the exact expression of the number of Eulerian circuits for odd n is unknown (it is clear that EC(Kn) = 0 for even n) and only the asymptotic formula was obtained (see [10]): as