Choosability in signed planar graphs

TL;DR

This paper investigates the list coloring properties of signed planar graphs, establishing bounds on choosability and demonstrating differences from unsigned graphs through specific constructions.

Contribution

It proves that signed planar graphs are 5-choosable, identifies conditions for 4- and 3-choosability based on girth, and provides counterexamples highlighting differences from unsigned graphs.

Findings

Signed planar graphs are 5-choosable.

Certain girth conditions guarantee 4- or 3-choosability.

Counterexamples show signed graphs can be less choosable than unsigned ones.

Abstract

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each $k \in \{3,4,5,6\}$, every signed planar graph without circuits of length $k$ is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of length 4 is 3-choosable. We construct a signed planar graph with girth 4 which is not 3-choosable but the unsigned graph is 3-choosable.