A Remark on Triangle-Critical Graphs

TL;DR

This paper investigates a special class of graphs called triangle-critical graphs, proving that for chromatic number up to 6, the only such graphs are complete graphs, thus partially confirming Toft's conjecture.

Contribution

The paper proves that for triangle-critical graphs with chromatic number at most 6, the only examples are complete graphs, advancing understanding of their structure.

Findings

For k ≤ 6, only complete graphs are triangle-critical.

Method of M. Stiebitz applied to analyze triangle-critical graphs.

Partial confirmation of Toft's conjecture for small k.

Abstract

A connected $k$-chromatic graph $G$ with $k \geq 3$ is said to be triangle-critical, if every edge of $G$ is contained in an induced triangle of $G$ and the removal of any triangle from $G$ decreases the chromatic number of $G$ by three. B. Toft posed the problem of showing that the complete graphs on more than two vertices are the only triangle-critical graphs. By applying a method of M. Stiebitz [Discrete Math. 64 (1987), 91--93], we answer the problem affirmatively for triangle-critical $k$-chromatic graphs with $k \leq 6$.