Binomial Inequalities for Chromatic, Flow, and Tension Polynomials

TL;DR

This paper establishes binomial coefficient inequalities for chromatic, flow, and tension polynomials, advancing the understanding of their coefficient structures and relations to Ehrhart polynomials and unimodular triangulations.

Contribution

It introduces new binomial coefficient inequalities for these polynomials, linking them to Ehrhart and order polynomials, and extends prior algebraic-combinatorial results.

Findings

Proves inequalities like hi^*_j hi^*_{d-j} for chromatic polynomials.

Extends inequalities to flow and tension polynomials.

Connects polynomial inequalities to Ehrhart theory and unimodular triangulations.

Abstract

A famous and wide-open problem, going back to at least the early 1970's, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute such inequalities when a chromatic polynomial $\chi_G(n) = \chi^*_0 \binom {n+d} d + \chi^*_1 \binom {n+d-1} d + \dots + \chi^*_d \binom n d$ is written in terms of a binomial-coefficient basis. For example, we show that $\chi^*_{ j } \le \chi^*_{ d-j }$, for $0 \le j \le \frac{ d }{ 2 }$. Similar results hold for flow and tension polynomials enumerating either modular or integral nowhere-zero flows/tensions of a graph. Our theorems follow from connections among chromatic, flow, tension, and order polynomials, as well as Ehrhart polynomials of lattice polytopes that admit unimodular triangulations. Our results use Ehrhart inequalities due to Athanasiadis and Stapledon and are related to recent work by Hersh--Swartz and Breuer--Dall, where inequalities similar to some of ours were derived using algebraic-combinatorial methods.