On Fair and Tolerant Colorings of Graphs

TL;DR

This paper investigates the concept of fair and tolerant vertex colorings in graphs, introducing the FAT chromatic number, and proves that certain bounds relating it to the traditional chromatic number do not exist.

Contribution

The paper establishes that no universal functions can bound the FAT chromatic number in terms of the traditional chromatic number, answering open questions negatively.

Findings

Both bounding functions do not exist for all graphs.

Negative resolution to open questions about FAT chromatic number.

Introduces and studies the FAT chromatic number concept.

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 $\alpha$ 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.