A Remark on Triangle-Critical Graphs

A connected $k$-chromatic graph $G$ with $k\ge 3$ 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 $k\le 6$.