A character approach to directed genus distribution of graphs: the bipartite single-black-vertex case

TL;DR

This paper proves the log-concavity of genus distribution in certain Eulerian digraphs using algebraic and combinatorial methods, extending results to specific factorizations in symmetric groups.

Contribution

It introduces a novel approach combining real-rooted polynomials and symmetric group representation theory to analyze genus distributions.

Findings

Genus distribution is log-concave for two families of Eulerian digraphs.

The method extends to factorizations of the identity in symmetric groups.

Provides positive answers to a question posed in 2002.

Abstract

Given an Eulerian digraph, we consider the genus distribution of its face-oriented embeddings. We prove that such distribution is log-concave for two families of Eulerian digraphs, thus giving a positive answer for these families to a question asked in Bonnington, Conder, Morton and McKenna (2002). Our proof uses real-rooted polynomials and the representation theory of the symmetric group $\mathbb{S}_n$. The result is also extended to some factorizations of the identity in $\mathbb{S}_n$ that are rotation systems of some families of one-face constellations.