Fair and Tolerant (FAT) Graph Colorings

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 ${\chi }^{\mathrm{F}\mathrm{A}\mathrm{T}}(G)$ as the largest integer $k$ for which $G$ admits a FAT $k$-coloring. We establish general bounds on ${\chi }^{\mathrm{F}\mathrm{A}\mathrm{T}}$, 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.