Vizing-Goldberg type bounds for the equitable chromatic number of block graphs

TL;DR

This paper proposes a conjecture providing a tight bound on the equitable chromatic number of block graphs, supported by verification in specific subclasses and structural characterizations.

Contribution

It introduces a conjecture analogous to Vizing-Goldberg bounds for equitable coloring of block graphs and verifies it in several important subclasses.

Findings

Conjecture suggests a gap-one bound for equitable chromatic number.

Verified the conjecture for well-covered block graphs.

Confirmed the conjecture for block graphs with specific structural properties.

Abstract

An equitable coloring of a graph $G$ is a proper vertex coloring of $G$ such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Goldberg conjecture for multigraphs. The results obtained in the paper support our conjecture. More precisely, we verify it in the class of well-covered block graphs, which are block graphs in which each vertex belongs to a maximum independent set. We also show that the conjecture is true for block graphs, which contain a vertex that does not lie in an independent set of size larger than two. Finally, we verify the conjecture for some symmetric-like block graphs. In order to derive our results we obtain structural characterizations of block graphs from these classes.