Broken-Cycle-Free Subgraphs and the Log-Concavity Conjecture for Chromatic Polynomials

Abstract

This paper concerns the coefficients of the chromatic polynomial of a graph. We first report on a computational verification of the strict log-concavity conjecture for chromatic polynomials for all graphs on at most $11$ vertices, as well as for certain cubic graphs.

In the second part of the paper we give a number of conjectures and theorems regarding the behavior of the coefficients of the chromatic polynomial, in part motivated by our computations. Here our focus is on $\varepsilon(G)$, the average size of a broken-cycle-free subgraph of the graph $G$, whose behavior under edge deletion and contraction is studied.