Proportional Choosability: A New List Analogue of Equitable Coloring

Abstract: In 2003, Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. In this paper, we motivate and define a new list analogue of equitable coloring called proportional choosability. A $k$-assignment $L$ for a graph $G$ specifies a list $L(v)$ of $k$ available colors for each vertex $v$ of $G$. An $L$-coloring assigns a color to each vertex $v$ from its list $L(v)$. For each color $c$, let $\eta(c)$ be the number of vertices $v$ whose list $L(v)$ contains $c$. 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. A graph $G$ is proportionally $k$-choosable if a proportional $L$-coloring of $G$ exists whenever $L$ is a $k$-assignment for $G$. We show that if a graph $G$ is proportionally $k$-choosable, then every subgraph of $G$ is also proportionally $k$-choosable and also $G$ is proportionally $(k+1)$-choosable, unlike equitable choosability for which analogous claims would be false. We also show that any graph $G$ is proportionally $k$-choosable whenever $k \geq \Delta(G) + \lceil |V(G)|/2 \rceil$, and we use matching theory to completely characterize the proportional choosability of stars and the disjoint union of cliques.