Binomial Inequalities for Chromatic, Flow, and Tension Polynomials

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