Graph theoretic and algorithmic aspect of the equitable coloring problem in block graphs

TL;DR

This paper investigates the equitable coloring problem in block graphs, exploring graph theoretic properties and algorithmic implications, particularly focusing on fixed-parameter tractability despite the problem's computational hardness.

Contribution

It provides new graph theoretic insights and algorithmic results, including fixed-parameter tractability analysis for equitable coloring in block graphs.

Findings

Equitable coloring remains hard in block graphs.

Graph parameters influence equitable coloring properties.

Fixed-parameter algorithms are developed for the problem.

Abstract

An equitable coloring of a graph $G=(V,E)$ is a (proper) vertex-coloring of $G$, such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. Then we use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.