Concerning FAT Colorings of Graphs

TL;DR

This paper proves the existence of an infinite sequence of regular graphs with positive degree that admit specific FAT colorings, answering an open question about their existence.

Contribution

It explicitly constructs infinitely many non-homomorphically equivalent regular graphs that admit FAT colorings with given parameters, confirming their existence.

Findings

Constructed an infinite sequence of such graphs.

Confirmed the existence of FAT colorings with specified parameters.

Provided explicit examples of non-homomorphically equivalent graphs.

Abstract

Let $G$ be a graph and let $C$ be a color set of cardinality $k$. Suppose $c \colon V(G) \to C$ is a (not necessarily proper) vertex coloring whose all color classes are $V_1$, $V_2$, $\dots$, $V_k$, each of which is nonempty. The vertex coloring $c$ is said to be a {\it FAT $k$-coloring of $G$} if there exist real numbers $\alpha$ and $\beta$, both in $[0,1]$, such that for every vertex $v\in V(G)$ and every color class $V_i$ the following equalities hold: $$ \bigl| V_i \cap N(v) \bigr| = \begin{cases} \alpha \deg (v) & \mbox{ if } \ \ v \notin V_i \beta \deg (v) & \mbox{ if } \ \ v \in V_i . \end{cases} $$ Let $k > 1$ be a fixed integer, and let $\alpha \in \left[ 0 , \frac{1}{k-1} \right) \cap \mathbb{Q}$ and $\beta \in [ 0 , 1 ] \cap \mathbb{Q}$ be some fixed rational numbers satisfying $ \beta + (k-1) \alpha = 1 $. It was asked for the existence of a graph $G$ with $\delta (G) > 0$ admitting some FAT $k$-coloring with the corresponding parameters $\alpha$ and $\beta$. This paper settles the question in the affirmative. We explicitly construct a sequence $\displaystyle\left\{G_n\right\}_{n=1}^{\infty}$ of pairwise non-homomorphically equivalent graphs, each being a regular graph of positive degree, admitting a FAT $k$-coloring with the corresponding parameters $\alpha$ and $\beta$.