, Volume 33, Issue 4, pp 495–512 | Cite as

Critical graphs without triangles: An optimum density construction

  • Wesley Pegden
Original Paper


We 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 ≤ for any 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).

Mathematics Subject Classification (2010)

05C15 05C35 


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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematics Courant InstituteNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsRutgers University (New Brunswick)PiscatawayUSA

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