A density bound for triangle-free $4$-critical graphs

TL;DR

This paper establishes a lower bound on the edge count of triangle-free 4-critical graphs, providing a unified proof for several 3-colorability results in planar and surface-embedded graphs.

Contribution

It introduces a new density bound for triangle-free 4-critical graphs, unifying multiple known 3-colorability results and nearly achieving optimality.

Findings

Proves $e(G) \,\geq\, \frac{5v(G)+2}{3}$ for triangle-free 4-critical graphs.

Unifies proofs of 3-colorability for various classes of graphs.

Constructs examples close to the bound, showing near optimality.

Abstract

We prove that every triangle-free $4$-critical graph $G$ satisfies $e(G) \geq \frac{5v(G)+2}{3}$. This result gives a unified proof that triangle-free planar graphs are $3$-colourable, and that graphs of girth at least five which embed in either the projective plane, torus, or Klein Bottle are $3$-colourable, which are results of Gr\"{o}tzsch, Thomassen, and Thomas and Walls. Our result is nearly best possible, as Davies has constructed triangle-free $4$-critical graphs $G$ such that $e(G) = \frac{5v(G) + 4}{3}$. To prove this result, we prove a more general result characterizing sparse $4$-critical graphs with few vertex-disjoint triangles.