Planar 4-critical graphs with four triangles

TL;DR

This paper characterizes all planar 4-critical graphs with exactly four triangles, extending understanding of graph colorability and answering a longstanding question posed by Erdős in 1990.

Contribution

It provides a complete description of all planar 4-critical graphs with four triangles, filling a gap in graph coloring theory.

Findings

All such graphs are explicitly described.

The result extends the Grunbaum-Aksenov Theorem.

Answers Erdős's 1990 question.

Abstract

By the Grunbaum-Aksenov Theorem (extending Grotzsch's Theorem) every planar graph with at most three triangles is 3-colorable. However, there are infinitely many planar 4-critical graphs with exactly four triangles. We describe all such graphs. This answers a question of Erdos from 1990.