On the Equitable Choosability of the Disjoint Union of Stars

TL;DR

This paper investigates the equitable list coloring of disjoint unions of star graphs, establishing NP-completeness results and characterizing equitable 2-choosability for specific cases, advancing understanding of this graph coloring variant.

Contribution

It provides complexity results and complete characterizations for equitable 2-choosability of unions of stars, a problem previously not fully understood.

Findings

Determining equitable choosability of unions of stars is NP-complete with uniform lists.

Complete characterization of equitable 2-choosability for two stars and identical star unions.

Results on equitable k-choosability for unions of two stars for arbitrary k.

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 $m \in \mathbb{N}$ 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 $\sum_{i=1}^n K_{1,m_i}$, the disjoint union of $n$ stars. We show that determining whether $\sum_{i=1}^n K_{1,m_i}$ 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 $n \geq 2$ identical stars) is equitably 2-choosable, and we present results on the equitable $k$-choosability of the disjoint union of two stars for arbitrary $k$.