Complexity of choosability with a small palette of colors

TL;DR

This paper investigates the computational complexity of graph choosability with a limited palette of colors, revealing surprising differences from the traditional case and identifying certain graph classes that are always choosable.

Contribution

It extends known results on graph choosability to scenarios with a small, fixed number of colors and characterizes classes of graphs that are always choosable under these constraints.

Findings

Complexity results differ from the classical case with implicit color counts.

Identification of graph classes defined by block properties that are always choosable.

Extension of known choosability results to small color palettes.

Abstract

A graph is $\ell$-choosable if, for any choice of lists of $\ell$ colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs when the number $k$ of colors is limited. We get results which differ surprisingly from the usual case where $k$ is implicit and which extend known results for the usual case. We also exhibit some classes of graphs (defined by structural properties of their blocks) which are choosable.