Proportional 2-Choosability with a Bounded Palette

Abstract

Proportional choosability is a list coloring analogue of equitable coloring. Specifically, a k-assignment L for a graph G associates a list L(v) of k available colors to each $v \in V(G)$. An L-coloring assigns a color to each vertex v from its list L(v). A proportional L-coloring of G is a proper L-coloring in which each color $c \in \bigcup _{v \in V(G)} L(v)$ is used $\lfloor \eta (c)/k \rfloor$ or $\lceil \eta (c)/k \rceil$ times where $\eta (c)=\left|{\{v \in V(G) : c \in L(v) \}}\right|$. A graph G is proportionally k-choosable if a proportional L-coloring of G exists whenever L is a k-assignment for G. Motivated by earlier work, we initiate the study of proportional choosability with a bounded palette by studying proportional 2-choosability with a bounded palette. In particular, when $\ell \ge 2$, a graph G is said to be proportionally $(2, \ell )$-choosable if a proportional L-coloring of G exists whenever L is a 2-assignment for G satisfying $|\bigcup _{v \in V(G)} L(v)| \le \ell$. We observe that a graph is proportionally (2, 2)-choosable if and only if it is equitably 2-colorable. As $\ell$ gets larger, the set of proportionally $(2, \ell )$-choosable graphs gets smaller. We show that whenever $\ell \ge 5$ a graph is proportionally $(2, \ell )$-choosable if and only if it is proportionally 2-choosable.