Induced subgraphs of graphs with large chromatic number. II. Three steps towards Gyárfás' conjectures

Gyarfas conjectured in 1985 that for all k, ?, every graph with no clique of size more than k and no odd hole of length more than ? has chromatic number bounded by a function of k, ?. We prove three weaker statements:Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five;For all ?, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than ?;For all k, ?, every graph with no clique of size more than k and sufficiently large chromatic number contains either a 5-hole or a hole of length more than ?.

[1]  Nicolas Trotignon,et al.  On graphs with no induced subdivision of K4 , 2012, J. Comb. Theory, Ser. B.

[2]  Nicolas Bousquet,et al.  Scott's Induced Subdivision Conjecture for Maximal Triangle-Free Graphs , 2011, Combinatorics, Probability and Computing.

[3]  Paul Seymour,et al.  Bounding the vertex cover number of a hypergraph , 1994, Comb..

[4]  Alex D. Scott.  Induced Cycles and Chromatic Number , 1999, J. Comb. Theory, Ser. B.

[5]  Paul D. Seymour,et al.  Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes , 2015, Journal of combinatorial theory. Series B (Print).

[6]  Paul D. Seymour,et al.  Induced subgraphs of graphs with large chromatic number. I. Odd holes , 2014, J. Comb. Theory, Ser. B.

[7]  Alex Scott.  Induced trees in graphs of large chromatic number , 1997 .

[8]  William T. Trotter,et al.  Triangle-free intersection graphs of line segments with large chromatic number , 2012, J. Comb. Theory, Ser. B.

[9]  Maria Chudnovsky,et al.  Colouring graphs with no long holes , 2015 .

[10]  Paul D. Seymour,et al.  Induced Subgraphs of Graphs with Large Chromatic Number. III. Long Holes , 2015, Comb..

[11]  A. Gyárfás.  Problems from the world surrounding perfect graphs , 1987 .

[12]  R. Graham,et al.  Ramsey theory (2nd ed.) , 1990 .

[13]  Paul D. Seymour,et al.  Induced Subgraphs of Graphs with Large Chromatic Number IX: Rainbow Paths , 2017, Electron. J. Comb..

[14]  Sean McGuinness,et al.  Colouring Arcwise Connected Sets in the Plane II , 2000, Graphs Comb..