Number of Eulerian orientations for Benjamini--Schramm convergent graph sequences

TL;DR

This paper proves that for sequences of Eulerian graphs converging in the Benjamini--Schramm sense, the normalized logarithm of the number of Eulerian orientations converges, revealing a form of asymptotic stability.

Contribution

It establishes the convergence of the normalized logarithm of Eulerian orientations for Benjamini--Schramm convergent graph sequences, a new result linking graph limits and Eulerian orientations.

Findings

Normalized log of Eulerian orientations converges for graph sequences

Benjamini--Schramm convergence implies asymptotic stability of Eulerian orientations

Provides a new connection between graph limits and combinatorial enumeration

Abstract

For a graph $G$ let $\varepsilon(G)$ denote the number of Eulerian orientations, and $v(G)$ denote the number of vertices of $G$. We show that if $(G_n)_n$ is a sequence of Eulerian graphs that are convergent in Benjamini--Schramm sense, then $\lim\limits_{n\to \infty}\frac{1}{v(G_n)}\ln \varepsilon(G_n)$ is convergent.