On Fair and Tolerant Colorings of Graphs
Saeed Shaebani
TL;DR
This work investigates Fair And Tolerant (FAT) colorings, showing that the FAT chromatic number $χ^{\rm FAT}(G)$ can differ drastically from the ordinary chromatic number $χ(G)$ in both directions. By constructing explicit graphs—disconnected graphs built from disjoint cliques and connected graphs based on near-complete bipartite structures and pendant triangles—the authors demonstrate unbounded gaps: for any $L_1<L_2$, there exist graphs with $χ(G)=L_1$ and $χ^{\rm FAT}(G)=L_2$, and for any $1<L_1<L_2$, graphs with $χ^{\rm FAT}(G)=L_1$ and $χ(G)=L_2$. In the connected case, they show the FAT number can be arbitrarily large while the ordinary chromatic number stays fixed at 2, and vice versa, establishing that FAT colorings can behave very differently from standard colorings even within connected graphs. The paper also proves the strongest possible general results: for every $n\ge3$ there are graphs with $χ(G)=1$ and $χ^{\rm FAT}(G)=n$, and graphs with $χ(G)=n$ and $χ^{\rm FAT}(G)=1$, confirming unbounded gaps in both directions and answering the Beers–Mulas questions negatively or positively as appropriate. Overall, the results underscore the surprising flexibility and extremal potential of FAT colorings across graph classes.
Abstract
A (not necessarily proper) vertex coloring of a graph $G$ with color classes $V_1$, $V_2$, $\dots$, $V_k$, is said to be a {\it Fair And Tolerant vertex coloring of $G$ with $k$ colors}, whenever $V_1$, $V_2$, $\dots$, $V_k$ are nonempty and there exist two real numbers $α$ and $β$ such that $α\in [0,1]$ and $β\in [0,1]$ and the following condition holds for each arbitrary vertex $v$ and every arbitrary color class $V_i$: $$ \bigl| V_i \cap N (v) \bigr| = \begin{cases} α°(v) & \mbox{ if } \ \ v \notin V_i β°(v) & \mbox{ if } \ \ v \in V_i . \end{cases} $$ The {\it FAT chromatic number} of $G$, denoted by $χ^{\rm FAT} (G)$, is defined as the maximum positive integer $k$ for which $G$ admits a Fair And Tolerant vertex coloring with $k$ colors. The concept of the FAT chromatic number of graphs was introduced and studied by Beers and Mulas, where they asked for the existence of a function $f \colon \mathbb{N} \to \mathbb{R}$ in such a way that the inequality $χ^{\rm FAT} (G) \ \leq \ f \big( χ(G) \big)$ holds for all graphs $G$. Another similar interesting question concerns the existence of some function $g \colon \mathbb{N} \to \mathbb{R}$ such that the inequality $χ(G) \ \leq \ g \left( χ^{\rm FAT} (G) \right)$ holds for every graph $G$. In this paper, we establish that both questions admit negative resolutions.