{"paper":{"title":"Critical graphs without triangles: an optimum density construction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wesley Pegden","submitted_at":"2011-01-24T00:29:54Z","abstract_excerpt":"We construct dense, triangle-free, chromatic-critical graphs of chromatic number $k$ for all $k\\geq 4$. For $k\\geq 6$ our constructions have $> (\\frac{1}{4} -\\varepsilon)n^2$ edges, which is asymptotically best possible by Tur\\'an's theorem. We also demonstrate (nonconstructively) the existence of dense $k$-critical graphs avoiding all odd cycles of length $\\leq \\ell$ for any $\\ell$ and any $k\\geq 4$, again with a best possible density of $>(\\frac{1}{4} -\\varepsilon)n^2$ edges for $k\\geq 6$. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4417","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}