Polynomials Counting Group Colorings in Graphs

TL;DR

This paper introduces a polynomial framework for counting group colorings and tensions in graphs, extending previous work to include cycle-assigning polynomials and their combinatorial interpretations.

Contribution

It defines the cycle-assigning polynomial for graphs, linking it to group colorings, tensions, and broken cycle concepts, thus unifying various combinatorial enumeration methods.

Findings

The polynomial counts $(A,f)$-colorings for Abelian groups.

Coefficients relate to counts of $ ext{ extit{alpha}}$-compatible spanning subgraphs.

Coefficients exhibit alternating signs and are nonzero under certain conditions.

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 $\alpha$-compatible graph and the cycle-assigning polynomial $P(G, \alpha; k)$ at $k$ in terms of $\alpha$-compatible spanning subgraphs, where $\alpha$ is an assigning of $G$ from its cycles to $\{0,1\}$. We prove that $P(G,\alpha;k)$ 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 $\alpha_{D,f}$ given by $f$ equals $\alpha$. Such an assigning is admissible. Based on Kochol's work, we derive that $k^{-c(G)}P(G,\alpha;k)$ 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 $P(G,\alpha;k)$ associated with admissible assignings $\alpha$ equals the number of $\alpha$-compatible spanning subgraphs that have $i$ edges and contain no broken $\alpha$-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 $\alpha$ of $G$ with $\alpha(e)=1$ for every loop $e$, the coefficients of $P(G,\alpha;k)$ are nonzero and alternate in sign.