Critical graphs without triangles: An optimum density construction

AbstractWe construct dense, triangle-free, chromatic-critical graphs of chromatic number k for all k ≥ 4. For k ≥ 6 our constructions have $$> \left( {\tfrac{1} {4} - \varepsilon } \right)n^2$$ edges, which is asymptotically best possible by Turán’s theorem. We also demonstrate (nonconstructively) the existence of dense k-critical graphs avoiding all odd cycles of length ≤ ℓ for any ℓ and any k≥4, again with a best possible density of $$> \left( {\tfrac{1} {4} - \varepsilon } \right)n^2$$ edges for k ≥ 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 ≥ 6).