Polynomials counting group colorings 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, $f:E\left(G\right)\to A$ and $D$ an orientation of a graph $G$. A vertex coloring $c:V\left(G\right)\to A$ is an $(A,f)$-coloring if $c\left(v\right)-c\left(u\right)\ne f\left(e\right)$ 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 Kochol (2022), and later extended the approach to regular matroids in Kochol (2024). 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 $f:E\left(G\right)\to A$ 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 $k^{-c\left(G\right)}P(G,\alpha ;k)$ 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 $k^{\left|V\left(G\right)\right|-i}$ 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 $\alpha \left(e\right)=1$ for every loop $e$, the coefficients of $P(G,\alpha ;k)$ are nonzero and alternate in sign.