Equitable coloring of corona products of cubic graphs is harder than ordinary coloring

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 \emph{equitable chromatic number} of $G$ and it is denoted by $\chi_{=}(G)$. In this paper the problem of determinig $\chi_=$ for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas of cubic graphs is solvable in polynomial time, the problem of equitable coloring becomes NP-hard for these graphs. We provide polynomially solvable cases of coronas of cubic graphs and prove the NP-hardness in a general case. As a by-product we obtain a simple linear time algorithm for equitable coloring of such graphs which uses $\chi_=(G)$ or $\chi_=(G)+1$ colors. Our algorithm is best possible, unless $P=NP$. Consequently, cubical coronas seem to be the only known class of graphs for which equitable coloring is harder than ordinary coloring.