Critical graphs without triangles: An optimum density construction

We construct dense, triangle-free, chromatic-critical graphs of chromatic number k for all k a parts per thousand yen 4. For k a parts per thousand yen 6 our constructions have edges, which is asymptotically best possible by Turan's theorem. We also demonstrate (nonconstructively) the existence of dense k-critical graphs avoiding all odd cycles of length a parts per thousand currency sign a"" for any a"" and any ka parts per thousand yen4, again with a best possible density of edges for k a parts per thousand yen 6. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the correct maximal density of k-critical members (k a parts per thousand yen 6).