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.

一个著名且广泛开放的问题，至少可以追溯到 1970 年代初期，它涉及图的色多项式的分类。对于这个分类问题，人们可能会要求色多项式的系数之间存在必要的不等式，并且当色多项式\(\chi _G(n)=\chi ^*_0\left( {\begin{array {c}n+d\\ d\end{array}}\right) +\chi ^*_1\left( {\begin{array}{c}n+d-1\\ d\end{array} }\right) +\dots +\chi ^*_d\left( {\begin{array}{c}n\\ d\end{array}}\right) \)以二项式系数为基础. 例如，我们证明\(\chi ^*_j\le \chi ^*_{dj}\)，对于\(0\le j\le d/2\). 类似的结果适用于枚举图形的模数或积分无处为零的流动/张力的流动和张力多项式。我们的定理来自色度、流动、张力和阶多项式之间的联系，以及允许单模三角剖分的格多面体的 Ehrhart 多项式。我们的结果使用了由 Athanasiadis 和 Stapledon 引起的 Ehrhart 不等式，并且与 Hersh-Swartz 和 Breuer-Dall 最近的工作有关，其中与我们的一些不等式相似的不等式是使用代数组合方法得出的。