Proportional Choosability: A New List Analogue of Equitable Coloring

TL;DR

This paper introduces proportional choosability, a new list coloring concept that generalizes equitable coloring, with properties like subgraph inheritance and bounds related to maximum degree and vertex count.

Contribution

It defines proportional choosability, proves its inheritance properties, establishes bounds based on maximum degree and vertex count, and characterizes it for stars and disjoint cliques.

Findings

Proportional choosability is inherited by subgraphs.

Every graph is proportionally k-choosable if k ≥ Δ(G) + ⌈|V(G)|/2⌉.

Complete characterization for stars and disjoint unions of cliques.

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.