On the Equitable Choosability of the Disjoint Union of Stars

Abstract

Equitable k-choosability is a list analogue of equitable k-coloring that was introduced by Kostochka, Pelsmajer, and West in 2003. It is known that if vertex disjoint graphs G 1 and G 2 are equitably k-choosable, the disjoint union of G 1 and G 2 may not be equitably k-choosable. Given any m ∈ N the values of k for which K 1,m is equitably k-choosable are known. Also, a complete characterization of equitably 2-choosable graphs is not known. With these facts in mind, we study the equitable choosability of n i=1 K 1,mi , the disjoint union of n stars. We show that determining whether n i=1 K 1,mi is equitably choosable is NP-complete when the same list of two colors is assigned to every vertex. We completely determine when the disjoint union of two stars (or n ≥ 2 identical stars) is equitably 2-choosable, and we present results on the equitable k-choosability of the disjoint union two stars for arbitrary k.