The number of Euler tours of a random $d$-in/$d$-out graph

Páidí Creed,Mary Cryan

doi:10.46298/dmtcs.2787

Abstract

In this paper we obtain the expectation and variance of the number of Euler tours of a random $d$-in/$d$-out directed graph, for $d \geq 2$. We use this to obtain the asymptotic distribution and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of a $d$-in/$d$-out graph is the product of the number of arborescences and the term $[(d-1)!]^n/n$. Therefore most of our effort is towards estimating the asymptotic distribution of the number of arborescences of a random $d$-in/$d$-out graph.

Highlights

P q Let G V, E be a directed graph
An Euler tour of G is any ordering eπp1q, . . . , eπp|E|q of the set of ¤ | | arcs E such that for every 1 i E, the target vertex of arc eπpiq is the source vertex of eπpi 1q, and p q such that the target vertex of eπp|E|q is the source of eπp1q
We use ET G to denote the set of Euler tours of G, where two Euler tours are considered to be equivalent if one is a cyclic permutation of the other

Summary

Introduction

P q Let G V, E be a directed graph. An Euler tour of G is any ordering eπp1q, . . . , eπp|E|q of the set of ¤ | | arcs E such that for every 1 i E , the target vertex of arc eπpiq is the source vertex of eπpi 1q, and p q such that the target vertex of eπp|E|q is the source of eπp1q. The simple algorithms for uniform sampling and approximate counting we consider, have so far only been analysed for Eulerian tournaments, in [9] (as part of their analysis of Euler tours on the undirected complete graph with an odd number of vertices). The proof idea we use in this paper is that of conditioning on short cycle counts, pioneered by Robinson and Wormald in [13, 14] Implicit in this pair of papers (and the subsequent work of Frieze et al [6]) is a characterisation of the asymptotic distribution of the number of Hamiltonian cycles in a random dregular graph in terms of random variables counting the number of i-cycles, for all fixed positive integers i. Some results for p q the more general model for in-degree/out-degree sequences d d1, . . . , dn , see [5]

Expectation and Variance of Euler tours

Asymptotic distribution of Euler tours

Generating and counting Euler tours