3-coloring triangle-free planar graphs with a precolored 9-cycle

TL;DR

This paper characterizes conditions under which a 3-coloring of a 9-cycle in a triangle-free planar graph cannot be extended to the entire graph, extending previous results for shorter cycles.

Contribution

It provides a complete characterization for 3-coloring extension in triangle-free planar graphs with a 9-cycle, advancing understanding beyond cycles of length 8.

Findings

Identifies specific configurations preventing 3-coloring extension

Extends known results from cycles of length up to 8 to length 9

Provides criteria for 3-coloring extension in the given class of graphs

Abstract

Given a triangle-free planar graph G and a 9-cycle C in G, we characterize situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results when C is a cycle of length at most 8.