Strong log-convexity of genus sequences

TL;DR

This paper disproves a longstanding conjecture that the sequence counting genus embeddings of a graph is log-concave, by providing counterexamples showing significant deviations from log-concavity.

Contribution

It provides the first known counterexamples to the conjecture, demonstrating that the genus sequence can strongly deviate from log-concavity.

Findings

Counterexamples show deviation from log-concavity in genus sequences

Disproves the conjecture by Gross, Robbins, and Tucker

Highlights complexity of genus embedding enumeration

Abstract

For a graph $G$, and a nonnegative integer $g$, let $a_g(G)$ be the number of $2$-cell embeddings of $G$ in an orientable surface of genus $g$ (counted up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins, and Tucker [Genus distributions for bouquets of circles, J. Combin. Theory Ser. B 47 (1989), 292-306] proposed a conjecture that the sequence $a_0(G),a_1(G),a_2(G),\dots$ is log-concave for every graph $G$. This conjecture is reminiscent to the Heron-Rota-Welsh Log Concavity Conjecture that was recently resolved in the affirmative by June Huh et al., except that it is closer to the notion of $\Delta$-matroids than to the usual matroids. In this short paper, we disprove the Log Concavity Conjecture of Gross, Robbins, and Tucker by providing examples that show strong deviation from log-concavity at multiple terms of their genus sequences.