Polynomials Counting Group Colorings in Graphs
Houshan Fu

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 be an additive Abelian group, and an orientation of a graph . A vertex coloring is an -coloring if for each oriented edge from to under . 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 -compatible graph and the cycle-assigning polynomial at in terms of -compatible spanning subgraphs, where is an assigning of from its cycles to . We prove that evaluates the number of -colorings of for any…
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Taxonomy
TopicsAdvanced Graph Theory Research
