Small subsets inherit sparse epsilon-regularity

In this paper we investigate the behaviour of subgraphs of sparse @e-regular bipartite graphs G=(V"1@?V"2,E) with vanishing density d that are induced by small subsets of vertices. In particular, we show that, with overwhelming probability, a random set S@?V"1 of size s@?1/d contains a subset S^' with |S^'|>=(1-@e^')|S| that induces together with V"2 an @e^'-regular bipartite graph of density (1+/-@e^')d, where @e^'->0 as @e->0. The necessity of passing to a subset S^' is demonstrated by a simple example. We give two applications of our methods and results. First, we show that, under a reasonable technical condition, ''robustly high-chromatic'' graphs contain small witnesses for their high chromatic number. Secondly, we give a structural result for almost all C"@?-free graphs on n vertices and m edges for odd @?, as long as m is not too small, and give some bounds on the number of such graphs for arbitrary @?.

[1]  Tibor Szabó,et al.  Turán's theorem in sparse random graphs , 2003, Random Struct. Algorithms.

[2]  E. Szemerédi.  Regular Partitions of Graphs , 1975 .

[3]  Yoshiharu Kohayakawa,et al.  Szemerédi’s Regularity Lemma and Quasi-randomness , 2003 .

[4]  Yoshiharu Kohayakawa,et al.  OnK4-free subgraphs of random graphs , 1997, Comb..

[5]  Jaroslav Nesetril,et al.  Surveys in Combinatorics: Partition theory and its applications , 1979 .

[6]  Stefanie Gerke,et al.  The sparse regularity lemma and its applications , 2005, BCC.

[7]  Stefanie Gerke,et al.  A probabilistic counting lemma for complete graphs , 2007 .

[8]  Y. Kohayakawa,et al.  Turán's extremal problem in random graphs: Forbidding odd cycles , 1996, Comb..

[9]  Vojtech Rödl.  On universality of graphs with uniformly distributed edges , 1986, Discret. Math..

[10]  János Komlós,et al.  The Regularity Lemma and Its Applications in Graph Theory , 2000, Theoretical Aspects of Computer Science.

[11]  Yoshiharu Kohayakawa,et al.  An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth , 1998, Comb..

[12]  Zoltán Füredi,et al.  Random Ramsey graphs for the four-cycle , 1994, Discret. Math..

[13]  Yoshiharu Kohayakawa,et al.  The Turán Theorem for Random Graphs , 2004, Comb. Probab. Comput..

[14]  Yoshiharu Kohayakawa,et al.  Threshold functions for asymmetric Ramsey properties involving cycles , 1997 .

[15]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[16]  Tomasz Łuczak,et al.  On triangle-free random graphs , 2000 .

[17]  V. Rödl,et al.  Arithmetic progressions of length three in subsets of a random set , 1996 .

[18]  M. Simonovits,et al.  Szemeredi''s Regularity Lemma and its applications in graph theory , 1995 .

[19]  Y. Kohayakawa.  Szemerédi's regularity lemma for sparse graphs , 1997 .

[20]  Anusch Taraz,et al.  K4-free subgraphs of random graphs revisited , 2007, Comb..

[21]  V. Rödl,et al.  On graphs with small subgraphs of large chromatic number , 1985, Graphs Comb..

[22]  Yoshiharu Kohayakawa,et al.  Turán's Extremal Problem in Random Graphs: Forbidding Even Cycles , 1995, J. Comb. Theory, Ser. B.

[23]  Stefanie Gerke,et al.  K5-free subgraphs of random graphs , 2004, Random Struct. Algorithms.

[24]  Yoshiharu Kohayakawa,et al.  Regular pairs in sparse random graphs I , 2003, Random Struct. Algorithms.

[25]  Vojtech Rödl,et al.  Large triangle-free subgraphs in graphs withoutK4 , 1986, Graphs Comb..