Binomial Inequalities for Chromatic, Flow, and Tension Polynomials
Abstract
A famous and wideopen 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 $\chi_G(n) = \chi^*_0 \binom {n+d} d + \chi^*_1 \binom {n+d1} d + \dots + \chi^*_d \binom n d$ is written in terms of a binomialcoefficient basis. For example, we show that $\chi^*_{ j } \le \chi^*_{ dj }$, for $0 \le j \le \frac{ d }{ 2 }$. Similar results hold for flow and tension polynomials enumerating either modular or integral nowherezero 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 HershSwartz and BreuerDall, where inequalities similar to some of ours were derived using algebraiccombinatorial methods.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1804.00208
 Bibcode:
 2018arXiv180400208B
 Keywords:

 Mathematics  Combinatorics;
 05C31;
 05A15;
 05C15;
 05C21;
 06A11;
 52B20
 EPrint:
 9 pages, to appear in Discrete &