The number of Euler tours of a random directed graph

TL;DR

This paper analyzes the expected number, variance, and distribution of Euler tours in random Eulerian directed graphs with fixed out-degree sequences, leading to efficient algorithms for sampling and counting Euler tours.

Contribution

It provides the first asymptotic distribution and concentration results for Euler tours in such graphs, enabling polynomial-time algorithms for sampling and counting.

Findings

Derived expectation and variance of Euler tours

Established asymptotic distribution and concentration results

Developed polynomial-time algorithms for sampling and counting

Abstract

In this paper we obtain the expectation and variance of the number of Euler tours of a random Eulerian directed graph with fixed out-degree sequence. We use this to obtain the asymptotic distribution of the number of Euler tours of a random $d$-in/$d$-out graph 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 an Eulerian directed graph with out-degree sequence $\mathbf{d}$ is the product of the number of arborescences and the term $\frac{1}{n}[\prod_{v \in V}(d_v-1)!]$. Therefore most of our effort is towards estimating the moments of the number of arborescences of a random graph with fixed out-degree sequence.