Fine structure of 4-critical triangle-free graphs I. Planar graphs with two triangles and 3-colorability of chains

TL;DR

This paper characterizes planar graphs with two triangles regarding 4-cycle precoloring extension, providing insights into the structure of 4-critical triangle-free graphs embedded in surfaces.

Contribution

It offers an exact characterization of when precolorings of 4-cycles do not extend in planar graphs with two triangles, advancing understanding of 3-colorability in such graphs.

Findings

Characterization of planar graphs with two triangles where 4-cycle precolorings do not extend

Solution to precoloring extension problem for separated 4-cycles in triangle-free planar graphs

Application of results to analyze 4-critical triangle-free graphs on surfaces

Abstract

Aksenov proved that in a planar graph G with at most one triangle, every precoloring of a 4-cycle can be extended to a 3-coloring of G. We give an exact characterization of planar graphs with two triangles in that some precoloring of a 4-cycle does not extend. We apply this characterization to solve the precoloring extension problem from two 4-cycles in a triangle-free planar graph in the case that the precolored 4-cycles are separated by many disjoint 4-cycles. The latter result is used in followup papers to give detailed information about the structure of 4-critical triangle-free graphs embedded in a fixed surface.