Fair and Tolerant (FAT) Graph Colorings

TL;DR

This paper introduces FAT colorings, a new graph coloring concept balancing tolerance and fairness, establishes bounds and properties, and characterizes it for specific graph families.

Contribution

It defines the FAT chromatic number, relates it to graph structure and spectra, and provides complete characterizations for certain graph classes.

Findings

Bounds on FAT chromatic number established

Relation between FAT coloring and graph spectral properties

Complete characterization for specific graph families

Abstract

We introduce and study Fair and Tolerant colorings (FAT colorings), where each vertex tolerates a given fraction of same-colored neighbors while fairness is preserved across the other coloring classes. Moreover, we define the FAT chromatic number $\chi^{\mathrm{FAT}}(G)$ as the largest integer $k$ for which $G$ admits a FAT $k$-coloring. We establish general bounds on $\chi^{\mathrm{FAT}}$, relate it to structural and spectral properties of graphs, and characterize it completely for several families of graphs. We conclude with a list of open questions that suggest future directions.