  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 $k \leq 6$.
