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

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.

19 pages, 4 figures, 2 tables, submitted

References in corpus (2)

• A unified approach to polynomial sequences with only real zeros
• A generalization of the quadrangulation relation to constellations and hypermaps