Covers, Orientations and Factors

Péter Csikvári,András Imolay

doi:10.37236/8767

Abstract

Given a graph $G$ with only even degrees, let $\varepsilon(G)$ denote the number of Eulerian orientations, and let $h(G)$ denote the number of half graphs, that is, subgraphs $F$ such that $d_F(v)=d_G(v)/2$ for each vertex $v$. Recently, Borbényi and Csikvári proved that $\varepsilon(G)\geq h(G)$ holds true for all Eulerian graphs, with equality if and only if $G$ is bipartite. In this paper we give a simple new proof of this fact, and we give identities and inequalities for the number of Eulerian orientations and half graphs of a $2$-cover of a graph $G$.

Highlights

In this paper we study the number of orientations and factors of Eulerian graphs
Recall that a graph G is Eulerian if every vertex of G has even degree
The best known result is due to Lieb [11] who determined the asymptotic number of Eulerian orientations of large grid graphs

Summary

Introduction

In this paper we study the number of orientations and factors of Eulerian graphs. Recall that a graph G is Eulerian if every vertex of G has even degree. Let h(G) denote the number of half graphs, that is, subgraphs F satisfying that dF (v) = dG(v)/2 for every vertex v. Borbenyi and Csikvari [4] proved that ε(G) h(G) holds true for all Eulerian graphs with equality if and and only if G is bipartite This inequality fits into a series of inequalities comparing the number of orientations and subgraphs with the same property. In this paper we give a simple new proof of the fact ε(G) h(G), and we give identities and inequalities for the number of Eulerian orientations and half graphs of a 2-cover of a graph G. For an edge set A ⊆ E(G) let ε(A) denote the number of Eulerian orientations of the graph (V, A).

Proof of Theorems 7 and 8

General 2-cover

Proof of Theorem 11