Equitable Colorings of Corona Multiproducts of Graphs

TL;DR

This paper investigates equitable colorings of corona products of graphs, providing new bounds and polynomial algorithms for specific cases, and confirming the Equitable Coloring Conjecture for these graph classes.

Contribution

It extends equitable coloring results to corona products, offering constructive polynomial algorithms and confirming the conjecture for these graph types.

Findings

Derived bounds for equitable chromatic number of corona products.

Provided polynomial algorithms for equitable coloring given initial colorings.

Confirmed the Equitable Coloring Conjecture for corona products of certain graphs.

Abstract

A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the equitable chromatic number of $G$ and denoted $\chi_{=}(G)$. It is known that this problem is NP-hard in general case and remains so for corona graphs. In "Equitable colorings of Cartesian products of graphs" (2012) Lin and Chang studied equitable coloring of Cartesian products of graphs. In this paper we consider the same model of coloring in the case of corona products of graphs. In particular, we obtain some results regarding the equitable chromatic number for $l$-corona product $G \circ ^l H$, where $G$ is an equitably 3- or 4-colorable graph and $H$ is an $r$-partite graph, a path, a cycle or a complete graph. Our proofs are constructive in that they lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of $G$. Moreover, we confirm Equitable Coloring Conjecture for corona products of such graphs. This paper extends our results from \cite{hf}.