
TL;DR
This paper introduces a weighted-set graph coloring problem with a polynomial that accounts for vertex color preferences, extending previous models and revealing new properties through analysis of various graph families.
Contribution
It develops a weighted-set chromatic polynomial for graph coloring with vertex preferences, generalizing prior work and analyzing its properties across different graph types.
Findings
Properties of the weighted-set chromatic polynomial are established.
Calculations for various graph families demonstrate new features.
The model extends previous unweighted coloring frameworks.
Abstract
We study a weighted-set graph coloring problem in which one assigns colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting that either disfavors or favors a given subset of colors contained in the set of colors. We construct and analyze a weighted-set chromatic polynomial associated with this coloring. General properties of this weighted-set chromatic polynomial are proved, and illustrative calculations are presented for various families of graphs. This study extends a previous one for the case and reveals a number of interesting new features.
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