Dense graphs with small clique number

We consider the structure of Kr-free graphs with large minimum degree, and show that such graphs with minimum degree δ>(2r − 5)n/(2r − 3) are homomorphic to the join Kr − 3∨H, where H is a triangle-free graph. In particular this allows us to generalize results from triangle-free graphs and show that Kr-free graphs with such a minimum degree have chromatic number at most r +1. We also consider the minimum-degree thresholds for related properties. Copyright © 2010 John Wiley & Sons, Ltd. 66:319-331, 2011

[1]  Richard H. Schelp,et al.  Nearly bipartite graphs , 2003, Discret. Math..

[2]  Guoping Jin,et al.  Triangle-Free Graphs with High Minimal Degrees , 1993, Combinatorics, Probability and Computing.

[3]  R. Shi.  The binding number of a graph and its pancyclism , 1987 .

[4]  S. Brandt,et al.  Subgraphs in vertex neighborhoods of K r -free graphs , 2004 .

[5]  Tomasz Luczak.  On The Structure Of Triangle-Free Graphs Of Large Minimum Degree , 2006, Comb..

[6]  Alexandr V. Kostochka,et al.  Graphs without short odd cycles are nearly bipartite , 1997, Discret. Math..

[7]  Carsten Thomassen,et al.  On the Chromatic Number of Triangle-Free Graphs of Large Minimum Degree , 2002, Comb..

[8]  Roland Häggkvist.  ODD Cycles of Specified Length in Non-Bipartite Graphs , 1982 .

[9]  S. Brandt.  Dense triangle-free graphs are four-colorable : A solution to the Erdős-Simonovits problem , 2005 .

[10]  C. C. Chen,et al.  Triangle-Free Graphs with Large Degree , 1997, Combinatorics, Probability and Computing.

[11]  Miklós Simonovits,et al.  On a valence problem in extremal graph theory , 1973, Discret. Math..

[12]  Noga Alon,et al.  H-Free Graphs of Large Minimum Degree , 2006, Electron. J. Comb..

[13]  Wayne Goddard,et al.  A note on maximal triangle-free graphs , 1993, J. Graph Theory.

[14]  Jørgen Bang-Jensen,et al.  Subgraphs in vertex neighborhoods of Kr-free graphs , 2004, J. Graph Theory.

[15]  Michael E. Orrison,et al.  The Linear Complexity of a Graph , 2006, Electron. J. Comb..

[16]  Guoping Jin,et al.  Odd Wheels in Graphs , 2002, J. Comb. Theory, Ser. B.

[17]  Tomasz Łuczak,et al.  On The Structure Of Triangle-Free Graphs Of Large Minimum Degree , 2006 .

[18]  Paul Erdös,et al.  On the connection between chromatic number, maximal clique and minimal degree of a graph , 1974, Discret. Math..

[19]  D. R. Woodall.  The binding number of a graph and its Anderson number , 1973 .