Weighted Graph Colorings

TL;DR

This paper introduces a weighted chromatic polynomial for graph coloring that accounts for vertex preferences, generalizes the classic chromatic polynomial, and provides structural formulas and zero analysis for specific graph families.

Contribution

It develops a new weighted chromatic polynomial, proves its properties, and demonstrates its ability to distinguish graphs beyond traditional chromatic polynomials.

Findings

Weighted chromatic polynomial generalizes the classic polynomial.

Structural formulas for lattice strip graphs are derived.

Zeros of the polynomial reveal complex graph coloring properties.

Abstract

We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We exhibit a weighted chromatic polynomial $Ph(G,q,w)$ associated with this problem that generalizes the chromatic polynomial $P(G,q)$. General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for $Ph(G,q,w)$ for lattice strip graphs $G$ with periodic longitudinal boundary conditions. The zeros of $Ph(G,q,w)$ in the $q$ and $w$ planes and their accumulation sets in the limit of infinitely many vertices of $G$ are analyzed. Finally, some related weighted graph coloring problems are mentioned.