On the Complexity of Local-Equitable Coloring in Claw-Free Graphs with Small Degree

Abstract

An equitable k-partition (\(k\ge 2\)) of a vertex set S is a partition of S into k subsets (may be empty sets) such that the sizes of any two subsets of S differ by at most one. A local-equitable k-coloring (\(k\ge 2\)) of G is an assignment of k colors to the vertices of G such that, for every maximal clique H of G, the coloring on H forms an equitable k-partition of H. Local-equitable coloring of graphs is a generalization of the proper vertex coloring of graphs and also a stronger version of clique-coloring of graphs. Claw-free graphs with maximum degree four are proved to be 2-clique-colorable [Discrete Math. Theoret. Comput. Sci. 11 (2) (2009), 15–24] but not necessary local-equitably 2-colorable. In this paper, given a claw-free graph G with maximum degree at most four, we present a linear time algorithm to give a local-equitable 2-coloring of G or decide that G is not local-equitably 2-colorable. As a corollary, we get that claw-free perfect graphs with maximum degree at most four are local-equitably 2-colorable.