On kk-chromatically connected graphs

A graph GG is chromatically kk-connected if every vertex cutset induces a subgraph with chromatic number at least kk. Thus, in particular each neighborhood has to induce akk-chromatic subgraph. Godsil, Nowakowski and Nešetřil asked whether there exists akk-chromatically connected graph such that every minimal cutset induces a subgraph with no triangles. We show that the answer is positive in a special case, when minimal is replaced by minimum cutsets. We will also answer another related question suggested by Nešetřil by proving the existence of highly chromatically connected graphs in which every vertex neighborhood induces a subgraph with a given girth.