On the Minimum Edge-Density of 4-Critical Graphs of Girth Five

TL;DR

This paper establishes a lower bound on the number of edges in 4-critical graphs with girth at least five and derives implications for graph colorability on certain surfaces.

Contribution

It proves a new edge-density bound for 4-critical graphs of girth five, extending previous results and applying a novel potential technique.

Findings

Lower bound |E(G)| >= (5|V(G)|+2)/3 for such graphs

Graphs embeddable in Klein bottle or torus with girth ≥ 5 are 3-colorable

Utilizes a new potential technique by Kostochka and Yancey

Abstract

We prove that if G is a 4-critical graph of girth at least five then |E(G)|>=(5|V(G)|+2)/3. As a corollary, graphs of girth at least five embeddable in the Klein bottle or torus are 3-colorable. These are results of Thomas and Walls, and Thomassen respectively. The proof uses the new potential technique developed by Kostochka and Yancey who proved that 4-critical graphs satisfy: |E(G)|>=(5|V(G)|-2)/3.