3-choosability of planar graphs with (<=4)-cycles far apart

TL;DR

This paper proves that planar graphs with small cycles (up to four) spaced far apart are 3-choosable, extending known results about 3-colorability to list coloring.

Contribution

It establishes a new sufficient condition for 3-choosability in planar graphs based on cycle distance constraints.

Findings

Planar graphs with cycles of length ≤4 far apart are 3-choosable.

The result parallels Havel's problem on 3-colorability with triangles.

Provides a new structural insight into list coloring of planar graphs.

Abstract

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. We prove that if cycles of length at most four in a planar graph G are pairwise far apart, then G is 3-choosable. This is analogous to the problem of Havel regarding 3-colorability of planar graphs with triangles far apart.