On Fair and Tolerant Colorings of Graphs

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 $\alpha$ and $\beta$ such that $\alpha \in [0,1]$ and $\beta \in [0,1]$ and the following condition holds for each arbitrary vertex $v$ and every arbitrary color class $V_i$: $$\bigl| V_i \cap N (v) \bigr| = \begin{cases} \alpha \deg (v) & \mbox{ if } \ \ v \notin V_i \beta \deg (v) & \mbox{ if } \ \ v \in V_i . \end{cases}$$ The {\it FAT chromatic number} of $G$, denoted by $\chi ^{{\rm FAT}} (G)$, 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 $\chi ^{{\rm FAT}} (G) \ \leq \ f \big( \chi (G) \big)$ 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 $\chi (G) \ \leq \ g \left( \chi ^{{\rm FAT}} (G) \right)$ holds for every graph $G$. In this paper, we establish that both questions admit negative resolutions.