A Simple Characterization of Proportionally 2-choosable Graphs

TL;DR

This paper characterizes graphs that are proportionally 2-choosable, showing they are linear forests with specific component size restrictions, and highlights open problems in equitable 2-choosability.

Contribution

It provides a complete characterization of proportionally 2-choosable graphs, a new concept related to equitable coloring, and discusses open problems in equitable 2-choosability.

Findings

Proportionally 2-choosable graphs are linear forests with a largest component of at most 5 vertices.

Other components in these graphs have at most two vertices.

Characterization of equitably 2-choosable graphs remains open.

Abstract

We recently introduced proportional choosability, a new list analogue of equitable coloring. Like equitable coloring, and unlike list equitable coloring (a.k.a. equitable choosability), proportional choosability bounds sizes of color classes both from above and from below. In this note, we show that a graph is proportionally 2-choosable if and only if it is a linear forest such that its largest component has at most 5 vertices and all of its other components have two or fewer vertices. We also construct examples that show that characterizing equitably 2-choosable graphs is still open.