Concerning FAT Colorings of 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 $\beta$.