3-coloring triangle-free planar graphs with a precolored 8-cycle

TL;DR

This paper characterizes the specific conditions under which a 3-coloring of an 8-cycle in a triangle-free planar graph cannot be extended to a proper 3-coloring of the entire graph, advancing understanding of graph coloring constraints.

Contribution

It provides a complete characterization of non-extendable 3-colorings for triangle-free planar graphs with an 8-cycle boundary, a novel result in graph coloring theory.

Findings

Identifies all configurations where 3-coloring extension fails.

Provides a characterization that can be used to determine extendability.

Enhances understanding of coloring properties in planar graphs.

Abstract

Let G be a planar triangle-free graph and let C be a cycle in G of length at most 8. We characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of the whole graph.