Note on 3-Choosability of Planar Graphs with Maximum Degree 4

TL;DR

This paper proves that a broad class of planar graphs with maximum degree 4, derived from bipartite plane graphs, are 3-choosable, expanding understanding beyond previously studied subclasses.

Contribution

It introduces a new class of planar graphs of maximum degree 4 that are proven to be 3-choosable, regardless of the presence of close triangles or short cycles.

Findings

Graphs are 3-choosable despite close triangles.

The class includes graphs with no forbidden short cycles.

Expands known subclasses of 3-choosable planar graphs.

Abstract

Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Gr\"{o}tzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prove that every planar graph of maximum degree $4$ obtained as a subgraph of the medial graph of any bipartite plane graph is $3$-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence representing an entirely different class than the graphs inferred by the above mentioned conjectures.