A note on the Turán function of even cycles

The Tur´an function ex(n, F) is the maximum number of edges in an F-free graph on n vertices. The question of estimating this function for F = C2k, the cycle of length 2k, is one of the central open questions in this area that goes back to the 1930s. We prove that ex(n,C2k) ≤ (k − 1) n1+1/k + 16(k − 1)n, improving the previously best known general upper bound of Verstra¨ete [Combin. Probab. Computing 9 (2000), 369–373] by a factor 8 + o(1) when n � k.

[1]  Felix Lazebnik,et al.  Polarities and 2k-cycle-free graphs , 1999, Discret. Math..

[2]  A. Rényii,et al.  ON A PROBLEM OF GRAPH THEORY , 1966 .

[3]  W. G. Brown.  On Graphs that do not Contain a Thomsen Graph , 1966, Canadian Mathematical Bulletin.

[4]  Keith E. Mellinger,et al.  LDPC Codes from Triangle-Free Line Sets , 2004, Des. Codes Cryptogr..

[5]  Jacques Verstraëte,et al.  On Arithmetic Progressions of Cycle Lengths in Graphs , 2000, Combinatorics, Probability and Computing.

[6]  Rephael Wenger,et al.  Extremal graphs with no C4's, C6's, or C10's , 1991, J. Comb. Theory, Ser. B.

[7]  Zoltán Füredi,et al.  On the Turán number for the hexagon , 2006 .

[8]  Christopher M. Hartman.  Extremal problems in graph theory , 1997 .

[9]  Zoltán Füredi,et al.  Graphs without quadrilaterals , 1983, J. Comb. Theory, Ser. B.

[10]  Miklós Simonovits,et al.  Compactness results in extremal graph theory , 1982, Comb..

[11]  Felix Lazebnik,et al.  Properties of Certain Families of 2k-Cycle-Free Graphs , 1994, J. Comb. Theory, Ser. B.

[12]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[13]  C. T. Benson.  Minimal Regular Graphs of Girths Eight and Twelve , 1966, Canadian Journal of Mathematics.

[14]  P. Erdös.  On an extremal problem in graph theory , 1970 .

[15]  M. Simonovits,et al.  Cycles of even length in graphs , 1974 .

[16]  Felix Lazebnik,et al.  Explicit Construction of Graphs with an Arbitrary Large Girth and of Large Size , 1995, Discret. Appl. Math..

[17]  Zoltán Füredi,et al.  On the Number of Edges of Quadrilateral-Free Graphs , 1996, J. Comb. Theory, Ser. B.

[18]  P. Erdös.  ON SEQUENCES OF INTEGERS NO ONE OF WHICH DIVIDES THE PRODUCT OF TWO OTHERS AND ON SOME RELATED PROBLEMS , 2004 .

[19]  Dhruv Mubayi,et al.  Constructions of bipartite graphs from finite geometries , 2005 .

[20]  C. R. J. Clapham,et al.  Graphs without four-cycles , 1989, J. Graph Theory.