Concerning FAT Colorings of Graphs

Paper

Abstract

Let

be a graph and let

be a color set of cardinality

. Suppose

is a (not necessarily proper) vertex coloring whose all color classes are

, each of which is nonempty. The vertex coloring

is said to be a {\it FAT

-coloring of

} if there exist real numbers

and

, both in

, such that for every vertex

and every color class

the following equalities hold:

Let

be a fixed integer, and let

and

be some fixed rational numbers satisfying

. It was asked for the existence of a graph

with

admitting some FAT

-coloring with the corresponding parameters

and

. This paper settles the question in the affirmative. We explicitly construct a sequence

of pairwise non-homomorphically equivalent graphs, each being a regular graph of positive degree, admitting a FAT

-coloring with the corresponding parameters

and