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$q$-Chromatic polynomials

Esme Bajo, Matthias Beck, Andrés R. Vindas-Meléndez

Abstract

We introduce and study a $q$-version of the chromatic polynomial of a given graph $G=(V,E)$, namely, \[ χ_G^λ(q,n) \ := \sum_{\substack{\text{proper colorings}\\ c\,:\,V\to[n]}} q^{ \sum_{ v \in V } λ_v c(v) }, \] where $λ\in \mathbb{Z}^V$ is a fixed linear form. Via work of Chapoton (2016) on $q$-Ehrhart polynomials, $χ_G^λ(q,n)$ turns out to be a polynomial in the $q$-integer $[n]_q$, with coefficients that are rational functions in $q$. Additionally, we prove structural results for $χ_G^λ(q,n)$ and exhibit connections to neighboring concepts, e.g., chromatic symmetric functions and the arithmetic of order polytopes. We offer a strengthened version of Stanley's conjecture that the chromatic symmetric function distinguishes trees, which leads to an analogue of $P$-partitions for graphs.

$q$-Chromatic polynomials

Abstract

We introduce and study a -version of the chromatic polynomial of a given graph , namely, \[ χ_G^λ(q,n) \ := \sum_{\substack{\text{proper colorings}\\ c\,:\,V\to[n]}} q^{ \sum_{ v \in V } λ_v c(v) }, \] where is a fixed linear form. Via work of Chapoton (2016) on -Ehrhart polynomials, turns out to be a polynomial in the -integer , with coefficients that are rational functions in . Additionally, we prove structural results for and exhibit connections to neighboring concepts, e.g., chromatic symmetric functions and the arithmetic of order polytopes. We offer a strengthened version of Stanley's conjecture that the chromatic symmetric function distinguishes trees, which leads to an analogue of -partitions for graphs.
Paper Structure (7 sections, 16 theorems, 72 equations, 1 figure)

This paper contains 7 sections, 16 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $q$-Ehrhart Polynomials
  3. The structure of $q$-chromatic polynomials
  4. The Leading Coefficient of a $q$-chromatic Polynomial
  5. $G$-partitions
  6. Open Questions
  7. Funding

Key Result

Theorem 1

There exists a unique polynomial $\widetilde{\chi}_G^\lambda(q,x)\in\mathbb{Q}(q)[x]$ such that where $[n]_q := \frac{ 1-q^n }{ 1-q }$.

Figures (1)

  • Figure 1: A table of the leading coefficients of the $q$-chromatic polynomials of the non-isomorphic trees on $d=6$ vertices.

Theorems & Definitions (37)

  • Theorem 1
  • Example 1
  • Conjecture 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Corollary 5
  • Corollary 6
  • ...and 27 more