The complexity of planar graph choosability

TL;DR

This paper investigates the computational complexity of determining k-choosability in planar graphs, proving NP-hardness for specific cases and providing explicit examples of non-choosable planar graphs.

Contribution

It establishes NP-hardness results for deciding 4-choosability in planar graphs and 3-choosability in planar triangle-free graphs, and constructs explicit non-choosable examples.

Findings

Deciding 4-choosability in planar graphs is NP-hard.

Deciding 3-choosability in planar triangle-free graphs is NP-hard.

Constructed explicit planar graphs that are not 4-choosable.

Abstract

A graph $G$ is {\em $k$-choosable} if for every assignment of a set $S(v)$ of $k$ colors to every vertex $v$ of $G$, there is a proper coloring of $G$ that assigns to each vertex $v$ a color from $S(v)$. We consider the complexity of deciding whether a given graph is $k$-choosable for some constant $k$. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable. We also obtain simple constructions of a planar graph which is not 4-choosable and a planar triangle-free graph which is not 3-choosable.