Fair and Tolerant (FAT) Graph Colorings
Lies Beers, Raffaella Mulas
TL;DR
The paper introduces Fair and Tolerant (FAT) colorings as a relaxation of proper graph colorings, formalizing a fixed tolerance parameter $\alpha$ with $\beta=1-(k-1)\alpha$ and defining the FAT chromatic number $\chi^{\mathrm{FAT}}(G)$. It derives fundamental structural properties, including a general bound $\chi^{\mathrm{FAT}}(G)\le \delta+1$ and, in regular connected graphs, equal-sized coloring classes and divisibility constraints, with $\beta=0$ recovering proper colorings. A central contribution is the spectral perspective: if a FAT $k$-coloring with parameter $\alpha$ exists, then $\lambda=k\alpha$ is an eigenvalue of the normalized Laplacian $L(G)$ with multiplicity at least $\max\{1,k-1\}$, yielding $\chi^{\mathrm{FAT}}(G)\le \mu+1$ where $\mu$ is the maximum multiplicity of $L(G)$; this is extended to regular graphs via $K$ and $A$ relationships. The authors completely characterize $\chi^{\mathrm{FAT}}$ for regular Turán graphs, show a merging framework for generating all FAT colorings from irreducibles, and identify irreducible colorings for several graph families, culminating in open questions about gaps with $\chi$, complexity, and extensions to other variants. These results connect combinatorial FAT colorings with spectral graph theory and provide a structured way to understand when fixed-fraction tolerance colorings are possible.
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 $χ^{\mathrm{FAT}}(G)$ as the largest integer $k$ for which $G$ admits a FAT $k$-coloring. We establish general bounds on $χ^{\mathrm{FAT}}$, 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.