A Remark on Triangle-Critical Graphs

Details A Remark on Triangle-Critical Graphs A Remark on Triangle-Critical Graphs

2008-02-24

arxiv.org/abs/0802.3529v1

Mathematics

Combinatorics

Scientific paper

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 $k \leq 6$.

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