A Note on the Equitable Choosability of Complete Bipartite Graphs

Jeffrey A. Mudrock, Madelynn Chase, Isaac Kadera, Ezekiel Thronburgh

Published 2018-08-04Version 1

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 $v \in V(G)$, and an equitable $L$-coloring of $G$ is a proper coloring, $f$, of $G$ such that $f(v) \in L(v)$ for each $v \in V(G)$ and each color class of $f$ has size at most $\lceil |V(G)|/k \rceil$. 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 that $K_{n,m}$ is equitably $k$-choosable if $k \geq \max \{n,m\}$. We prove $K_{n,m}$ is equitably $k$-choosable if $m \leq \left\lceil (m+n)/k \right \rceil(k-n)$ which gives $K_{n,m}$ is equitably $k$-choosable for certain $k$ satisfying $k < \max \{n,m\}$. We also give a complete characterization of the equitable choosability of stars (i.e. graphs of the form $K_{1,m}$).