Alon-Tarsi number of signed planar graphs

TL;DR

This paper extends the understanding of the Alon-Tarsi number for signed planar graphs, establishing upper bounds and providing examples related to colorability and choosability.

Contribution

It generalizes Zhu's result to signed graphs, showing the Alon-Tarsi number is at most 5, and proves a tighter bound of 4 for 2-colorable signed planar graphs.

Findings

Alon-Tarsi number of any signed planar graph is at most 5.

2-colorable signed planar graphs have Alon-Tarsi number at most 4.

Existence of 2-colorable signed planar graph not 3-choosable.

Abstract

Let $(G,\sigma)$ be any signed planar graph. We show that the Alon-Tarsi number of $(G,\sigma)$ is at most 5, generalizing a recent result of Zhu for unsigned case. In addition, if $(G,\sigma)$ is $2$-colorable then $(G,\sigma)$ has the Alon-Tarsi number at most 4. We also construct a signed planar graph which is $2$-colorable but not $3$-choosable.