Equitable List Coloring of Planar Graphs With Given Maximum Degree

Abstract

If $L$ is a list assignment of $r$ colors to each vertex of an $n$-vertex graph $G$, then an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/r\rceil$ times. A graph is equitably $r$-choosable if it has an equitable $L$-coloring for every $r$-list assignment $L$. In 2003, Kostochka, Pelsmajer, and West (KPW) conjectured that an analog of the famous Hajnal–Szemerédi Theorem on equitable coloring holds for equitable list coloring, namely, that for each positive integer $r$ every graph $G$ with maximum degree at most $r-1$ is equitably $r$-choosable. The main result of this paper is that for each $r\ge 9$ and each planar graph $G$, a stronger statement holds: if the maximum degree of $G$ is at most $r$, then $G$ is equitably $r$-choosable. In fact, we prove the result for a broader class of graphs—the class $ℬ$ of the graphs in which each bipartite subgraph $B$ with $\left|V\left(B\right)\right|\ge 3$ has at most $2\left|V\left(B\right)\right|-4$ edges. Together with some known results, this implies that the KPW Conjecture holds for all graphs in $ℬ$, in particular, for all planar graphs. We also introduce the new stronger notion of strongly equitable (SE, for short) list coloring and prove all bounds for this parameter. An advantage of this is that if a graph is SE $r$-choosable, then it is both equitably $r$-choosable and equitably $r$-colorable, while neither of being equitably $r$-choosable and equitably $r$-colorable implies the other.