Critical graphs without triangles: an optimum density construction

Details Critical graphs without triangles: an optimum density construction Critical graphs without triangles: an optimum density construction

2011-01-24

arxiv.org/abs/1101.4417v1

Mathematics

Combinatorics

17 pages, 5 figures, 1 table

Scientific paper

We construct dense, triangle-free, chromatic-critical graphs of chromatic number k for all k>/-4. For k>/-6 our constructions have >(1/4-{\espilon})n^2 edges, which is asymptotically best possible by Tur\'an's theorem. We also demonstrate (nonconstructively) the existence of dense pentagon-and-triangle-free k-critical graphs for any k>/-4, again with a best possible density of >(1/4-{\espilon})n^2 edges for k>/-6. The families of triangle-free and pentagon-and-triangle-free graphs are thus rare examples where we know the correct maximal density of k-critical members (k>/-6). We also construct dense 4-critical graphs of any odd-girth.

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