Binomial Inequalities for Chromatic, Flow, and Tension Polynomials

A famous and wide-open problem, going back to at least the early 1970s, 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 $$\chi _G(n)=\chi ^*_0\left( {\begin{array}{c}n+d\\ d\end{array}}\right) +\chi ^*_1\left( {\begin{array}{c}n+d-1\\ d\end{array}}\right) +\dots +\chi ^*_d\left( {\begin{array}{c}n\\ d\end{array}}\right) $$ is written in terms of a binomial-coefficient basis. For example, we show that $$\chi ^*_j\le \chi ^*_{d-j}$$ , for $$0\le j\le d/2$$ . 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.