The problem of finding dense induced bipartite subgraphs in H-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any H-free graph with minimum degree at least d contains an induced bipartite subgraph of minimum degree at least cH log d/log log d, thus nearly confirming one and proving another conjecture of Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdœs, Janson, Łuczak and Spencer.
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N. Alon: Explicit Ramsey graphs and orthonormal labelings, Elec. J. Combin.1 (1994), R12.
N. Alon: Bipartite subgraphs, Combinatorica16 (1996), no. 3, 301–311.
N. Alon, B. Bollobás, M. Krivelevich and B. Südakov: Maximum cuts and judicious partitions in graphs without short cycles, J. Combin. Theory Ser. B88 (2003), 329–346.
N. Alon, M. Krivelevich and B. Sudakov: Coloring graphs with sparse neighborhoods, J. Combin. Theory Ser. B77 (1999), 73–82.
N. Alon, M. Krivelevich and B. Sudakov: MaxCut in H-free graphs, Comb. Prob. Comput.14 (2005), 629–647.
B. Bollobás and A. D. Scott: Better bounds for max cut, in: Contemporary combinatorics, 185–246, Bolyai Soc. Math. Stud., 10, János Bolyai Math. Soc., Budapest, 2002.
W. Cames van Batenburg, R. de Joannis de Verclos, R. J. Kang and F. Pirot: Bipartite induced density in triangle-free graphs, arXiv:1808.02512.
D. Conlon, J. Fox, M. Kwan and B. Sudakov: Hypergraph cuts above the average, Israel J. Math., to appear.
P. Erdős: On some extremal problems in graph theory, Israel J. Math.3 (1965), 113–116.
P. Erdős: Problems and results in graph theory and combinatorial analysis, (1976), 169–192. Congressus Numerantium, No. XV.
C. S. Edwards: Some extremal properties of bipartite subgraphs, Canad. J. Math.3 (1973), 475–485.
P. Erdős, R. Faudree, J. Pach and J. Spencer: How to make a graph bipartite, J. Combin. Theory Ser. B45 (1988), 86–98.
P. Erdős and M. Simonovits: How many edges should be deleted to make a triangle-free graph bipartite?, in: Sets, graphs and numbers, Colloq. Math. Soc. János Bolyai 60, North-Holland, Amsterdam, 1992, 239–263.
P. Erdős, S. Janson, T. Łuczak and J. Spencer: A note on triangle-free graphs, Random discrete structures (Minneapolis, MN, 1993), IMA Vol. Math. Appl., vol. 76, Springer, New York, 1996, 117–119.
P. Erdős and M. Simonovits: Some extremal problems in graph theory, Combinatorial theory and its applications, I (Proc. Colloq., Balatonfüred, 1969), pp. 377–390, North-Holland, Amsterdam, 1970.
L. Esperet, R. J. Kang and S. Thomassé: Separation choosability and dense bipartite induced subgraphs, Comb. Prob. Comput., arXiv:1802.03727.
A. Frieze and M. Karoński: Introduction to random graphs, Cambridge University Press, 2016.
J. Gimbel and C. Thomassen: Coloring triangle-free graphs with fixed size, Discrete Math.219 (2000), no. 1–3, 275–277.
H. Guo and L. Warnke: Packing nearly optimal Ramsey R(3, t) graphs, Combinatorica, accepted.
D. G. Harris: Some results on chromatic number as a function of triangle count, SIAM J. Discrete Math.33 (2019), 546–563.
T. Jiang and R. Seiver: Turán numbers of subdivided graphs, SIAM J. Discrete Math.26 (2012), 1238–1255.
A. Johansson: Asymptotic choice number for triangle-free graphs, Technical Report 91-5, DIMACS, 1996.
M. Krivelevich: Bounding Ramsey numbers through large deviation inequalities, Random Struct. Algor.7 (1995), 145–155.
M. Krivelevich and B. Sudakov: Pseudo-random graphs, in More sets, graphs and numbers, 199–262, Springer, Berlin, Heidelberg, 2006.
M. Molloy: The list chromatic number of graphs with small clique number, J. Combin. Theory Ser. B134 (2019), 264–184.
A. Nilli: Triangle-free graphs with large chromatic numbers, Discrete Math.211 (2000), 261–262.
S. Poljak and Z. Tuza: Bipartite subgraphs of triangle-free graphs, SIAM J. Discrete Math.7 (1994), 307–313.
B. Sudakov: Making a K4-free graph bipartite, Combinatorica27 (2007), 509–518.
P. Turán: On an external problem in graph theory, Mat. Fiz. Lapok48 (1941), 436–452.
We would like to thank the anonymous referees for their helpful comments and suggestions.
Research supported in part by SNSF project 178493.
Research supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation.
Research supported in part by SNSF grant 200021-175573.
Research supported by the Alexander von Humboldt Foundation.
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Kwan, M., Letzter, S., Sudakov, B. et al. Dense Induced Bipartite Subgraphs in Triangle-Free Graphs. Combinatorica 40, 283–305 (2020). https://doi.org/10.1007/s00493-019-4086-0
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