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arXiv · 2510.18494

Fair and Tolerant (FAT) Graph Colorings

Lies Beers·Raffaella Mulas
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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.

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Lies Beers, Raffaella Mulas. 2025-11-13. Fair and Tolerant (FAT) Graph Colorings. https://arxiv.org/abs/2510.18494

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