Polynomials Counting Group Colorings in Graphs

TL;DR

The paper develops a cycle-assigning framework that generalizes graph colorings to Abelian-group colorings and tensions through a polynomial $P(G,\alpha;k)$ indexed by cycle-assignments $\alpha$. It establishes deletion-contraction recurrences, connects $P(G,\alpha;k)$ to Kochol’s nowhere-zero chain polynomials via regular matroids, and provides explicit expressions and decompositions (including $P=k^{c(G)}\tau$) that unify colorings, tensions, and chains. A generalized Whitney Broken Cycle Theorem counts coefficients of $P(G,\alpha;k)$ in terms of $\alpha$-compatible subgraphs avoiding broken cycles, and coefficients are shown to be order-preserving with respect to $\alpha$, leading to positivity and sign-structure results. The work also conjectures log-concavity/unimodality of the coefficients and explores implications for transferring colorability across Abelian groups of fixed order, offering a cohesive framework for group-coloring problems in graphs.

Abstract

Jaeger et al. in 1992 introduced group coloring as the dual concept to group connectivity in graphs. Let $A$ be an additive Abelian group, $ f: E(G)\to A$ and $D$ an orientation of a graph $G$. A vertex coloring $c:V(G)\to A$ is an $(A, f)$-coloring if $c(v)-c(u)\ne f(e)$ for each oriented edge $e=uv$ from $u$ to $v$ under $D$. Kochol recently introduced the assigning polynomial to count nowhere-zero chains in graphs--nonhomogeneous analogues of nowhere-zero flows in \cite{Kochol2022}, and later extended the approach to regular matroids in \cite{Kochol2024}. Motivated by Kochol's work, we define the $α$-compatible graph and the cycle-assigning polynomial $P(G, α; k)$ at $k$ in terms of $α$-compatible spanning subgraphs, where $α$ is an assigning of $G$ from its cycles to $\{0,1\}$. We prove that $P(G,α;k)$ evaluates the number of $(A,f)$-colorings of $G$ for any Abelian group $A$ of order $k$ and $f:E(G)\to A$ such that the assigning $α_{D,f}$ given by $f$ equals $α$. Such an assigning is admissible. Based on Kochol's work, we derive that $k^{-c(G)}P(G,α;k)$ is a polynomial enumerating $(A,f)$-tensions and counting specific nowhere-zero chains. Furthermore, by extending Whitney's broken cycle concept to broken compatible cycles, we show that the absolute value of the coefficient of $k^{|V(G)|-i}$ in $P(G,α;k)$ associated with admissible assignings $α$ equals the number of $α$-compatible spanning subgraphs that have $i$ edges and contain no broken $α$-compatible cycles. According to the combinatorial explanation, we establish a unified order-preserving relation from admissible assignings to cycle-assigning polynomials, and further show that for any admissible assigning $α$ of $G$ with $α(e)=1$ for every loop $e$, the coefficients of $P(G,α;k)$ are nonzero and alternate in sign.

Figures (1)

• Figure 1: The directed paths $P_1,P_2$ and $P_3$ toghter form an oriented theta-graph

Theorems & Definitions (37)

• Definition 1
• Definition 2
• Theorem 2.1: Counting Formula
• proof
• Theorem 2.2
• Corollary 1
• Lemma 1
• proof
• Theorem 2.3: Deletion-contraction Formula
• proof