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

TL;DR

This paper considers the equitable coloring problem in block graphs, and presents some graph theoretic results relating various parameters, mainly dealing with the fixed-parameter tractability of 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.

Figures (2)

• Figure 1: The construction of $G'$ from $G$ (cf. the proof of Observation \ref{['obs:Eqblockgraphswithperfectmatching']}): we add an edge with a clique of size $k$ to every vertex of $G$ not covered by the maxmimum matching. In this example, $k=3$.
• Figure 2: An exemplary block graph $G$ with $dc(G)-rad(G)=k$ with $k=7$.

Theorems & Definitions (19)

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