A Note on the Equitable Choosability of Complete Bipartite Graphs

• Main Authors: Mudrock Jeffrey A., Chase Madelynn, Thornburgh Ezekiel, Kadera Isaac, Wagstrom Tim
• Format: Article
• Language: English
• Published: Sciendo 2021-11-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: graph coloring; equitable coloring; list coloring; equitable choos-ability; 05c15
• Online Access: https://doi.org/10.7151/dmgt.2232

In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. A k-assignment, L, for a graph G assigns a list, L(v), of k available colors to each v ∈ V (G), and an equitable L-coloring of G is a proper coloring, f, of G such that f(v) ∈ L(v) for each v ∈ V (G) and each color class of f has size at most ⌈|V (G)|/k⌉. Graph G is said to be equitably k-choosable if an equitable L-coloring of G exists whenever L is a k-assignment for G. In this note we study the equitable choosability of complete bipartite graphs. A result of Kostochka, Pelsmajer, and West implies Kn,m is equitably k-choosable if k ≥ max{n, m} provided Kn,m ≠ K2l+1,2l+1. We prove Kn,m is equitably k-choosable if m ≤ ⌈ (m + n)/k⌉ (k − n) which gives Kn,m is equitably k-choosable for certain k satisfying k < max{n, m}. We also give a complete characterization of the equitable choosability of complete bipartite graphs that have a partite set of size at most 2.