Spanning 3-colourable subgraphs of small bandwidth in dense graphs

A conjecture by Bollobas and Komlos states the following: For every@c>0and integersr>=2and @D, there exists@b>0with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least(r-1r+@c)nand H is an r-chromatic graph with n vertices, bandwidth at most @bn and maximum degree at most @D, then G contains a copy of H. This conjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r=3.

[1]  Noga Alon,et al.  2-factors in Dense Graphs , 1996, Discret. Math..

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

[3]  Sarmad Abbasi.  How Tight Is the Bollobás-Komlós Conjecture? , 2000, Graphs Comb..

[4]  Endre Szemerédi,et al.  Proof of the Seymour conjecture for large graphs , 1998 .

[5]  Hal A. Kierstead,et al.  The Square of Paths and Cycles , 1995, J. Comb. Theory, Ser. B.

[6]  Daniela Kühn,et al.  Large planar subgraphs in dense graphs , 2005, J. Comb. Theory, Ser. B.

[7]  M. Aigner,et al.  Embedding Arbitrary Graphs of Maximum Degree Two , 1993 .

[8]  Noga Alon,et al.  AlmostH-factors in dense graphs , 1992, Graphs Comb..

[9]  Vojtech Rödl,et al.  Matchings Meeting Quotas and Their Impact on the Blow-Up Lemma , 2001, SIAM J. Comput..

[10]  Vojtech Rödl,et al.  Perfect Matchings in ε-Regular Graphs and the Blow-Up Lemma , 1999, Comb..

[11]  Ali Shokoufandeh,et al.  Proof of a tiling conjecture of Komlós , 2003, Random Struct. Algorithms.

[12]  Gábor N. Sárközy,et al.  An algorithmic version of the blow-up lemma , 1998 .

[13]  Claude Berge,et al.  The theory of graphs and its applications , 1962 .

[14]  P. Erdos,et al.  A LIMIT THEOREM IN GRAPH THEORY , 1966 .

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

[16]  Endre Szemerédi,et al.  Proof of a Conjecture of Bollobás and Eldridge for Graphs of Maximum Degree Three* , 2003, Comb..

[17]  János Komlós,et al.  On the square of a Hamiltonian cycle in dense graphs , 1996, Random Struct. Algorithms.

[18]  Victor Alexandrov,et al.  Problem section , 2007 .

[19]  János Komlós,et al.  Proof of the Alon-Yuster conjecture , 2001, Discret. Math..

[20]  Noga Alon,et al.  AlmostH-factors in dense graphs , 1992, Graphs Comb..

[21]  D. Osthus,et al.  Spanning triangulations in graphs , 2005 .

[22]  P. Erdos,et al.  On the maximal number of independent circuits in a graph , 1963 .

[23]  Gábor N. Sárközy,et al.  On the Pósa-Seymour conjecture , 1998 .

[24]  Frank Harary,et al.  Graph Theory , 2016 .

[25]  János Komlós,et al.  Tiling Turán Theorems , 2000, Comb..

[26]  Daniela Kühn,et al.  Critical chromatic number and the complexity of perfect packings in graphs , 2006, SODA '06.

[27]  G. Dirac.  Some Theorems on Abstract Graphs , 1952 .

[28]  Andrzej Czygrinow,et al.  2-factors in Dense Bipartite Graphs , 2002, Discret. Math..

[29]  János Komlós,et al.  Blow-up Lemma , 1997, Combinatorics, Probability and Computing.

[30]  Hal A. Kierstead,et al.  Hamiltonian Square-Paths , 1996, J. Comb. Theory, Ser. B.

[31]  Jfinos Koml s.  Proof of the Seymour Conjecture for Large Graphs , 2005 .

[32]  P. Erdös.  On the structure of linear graphs , 1946 .

[33]  Vojtech Rödl,et al.  The Algorithmic Aspects of the Regularity Lemma , 1994, J. Algorithms.

[34]  János Komlós,et al.  Spanning Trees in Dense Graphs , 2001, Combinatorics, Probability and Computing.

[35]  Vojtech Rödl,et al.  The Ramsey number of a graph with bounded maximum degree , 1983, J. Comb. Theory, Ser. B.