Closed formulas for fractional chromatic polynomials of some common classes of graphs

TL;DR

This paper introduces closed formulas for fractional chromatic polynomials of specific graph classes, extending classical results to fractional colorings and providing new tools for graph coloring analysis.

Contribution

It derives closed formulas for fractional chromatic polynomials of complete graphs, trees, and forests, and generalizes the Fundamental Reduction Theorem for fractional colorings.

Findings

Closed formulas for fractional chromatic polynomials of complete graphs, trees, and forests.

A generalized Fundamental Reduction Theorem for fractional colorings.

Extension of classical chromatic polynomial results to fractional colorings.

Abstract

Chromatic polynomials have been studied extensively, giving us results such as the Fundamental Reduction Theorem and closed formulas for the chromatic polynomials of common classes of graphs. Though, none of those extend to the context of fractional colorings. We thus present closed formulas for the "fractional" chromatic polynomial - a function that counts the number of distinct $b$-fold $\lambda$-colorings on a given graph - of complete graphs, trees and forests, as well as a generalized form of the Fundamental Reduction Theorem.