Graph Polynomials and Group Coloring of Graphs

TL;DR

This paper extends the understanding of group colorings in graphs by using graph polynomials to show that certain classes of graphs have exponentially many colorings, generalizing previous planar graph results.

Contribution

It introduces a polynomial-based approach to prove that $K_5$-minor-free graphs are $ield$-$5$-choosable with many colorings, broadening prior planar graph results.

Findings

Every $K_5$-minor-free graph on $n$ vertices is $ield$-$5$-choosable for fields with at least 5 elements.

Such graphs have at least $5^{n/4}$ colorings for any list assignment.

The approach generalizes previous results from planar graphs to a wider class using graph polynomials.

Abstract

Let $\Gamma$ be an Abelian group and let $G$ be a simple graph. We say that $G$ is $\Gamma$-colorable if for some fixed orientation of $G$ and every edge labeling $\ell:E(G)\rightarrow \Gamma$, there exists a vertex coloring $c$ by the elements of $\Gamma$ such that $c(y)-c(x)\neq \ell(e)$, for every edge $e=xy$ (oriented from $x$ to $y$). Langhede and Thomassen proved recently that every planar graph on $n$ vertices has at least $2^{n/9}$ different $\mathbb{Z}_5$-colorings. By using a different approach based on graph polynomials, we extend this result to $K_5$-minor-free graphs in the more general setting of field coloring. More specifically, we prove that every such graph on $n$ vertices is $\mathbb{F}$-$5$-choosable, whenever $\mathbb{F}$ is an arbitrary field with at least $5$ elements. Moreover, the number of colorings (for every list assignment) is at least $5^{n/4}$.