The CLLC conjecture holds for cyclic outer permutations

TL;DR

This paper proves the CLLC conjecture for cyclic permutations using combinatorial and algebraic methods, advancing understanding of log-concavity in graph genus distributions.

Contribution

It confirms the CLLC conjecture for cyclic permutations by employing Hultman numbers and the Hermite--Biehler theorem, suggesting all local genus polynomials may be real-rooted.

Findings

Confirmed CLLC conjecture for cyclic permutations

Applied Hermite--Biehler theorem to Stirling numbers

Proposed that all local genus polynomials are real-rooted

Abstract

Recently, Gross et al. posed the LLC conjecture for the locally log-concavity of the genus distribution of every graph, and provided an equivalent combinatorial version, the CLLC conjecture, on the log-concavity of the generating function counting cycles of some permutation compositions. In this paper, we confirm the CLLC conjecture for cyclic permutations, with the aid of Hultman numbers and by applying the Hermite--Biehler theorem on the generating function of Stirling numbers of the first kind. This leads to a further conjecture that every local genus polynomial is real-rooted.