Abstract
We show that every K 4-free graph G with n vertices can be made bipartite by deleting at most n 2/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős.
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References
N. Alon: Bipartite subgraphs, Combinatorica 16 (1996), 301–311.
N. Alon, B. Bollobás, M. Krivelevich and B. Sudakov: Maximum cuts and judicious partitions in graphs without short cycles, J. Combin. Theory Ser. B 88 (2003), 329–346.
N. Alon, M. Krivelevich and B. Sudakov: MaxCut in H-free graphs, Combinatorics, Probability and Computing 14 (2005), 629–647.
B. Bollobás and A. Scott: Better bounds for Max Cut, in: Contemporary combinatorics, Bolyai Soc. Math. Stud. 10, János Bolyai Math. Soc., Budapest, 2002, 185–246.
A. Bondy and S. Locke: Largest bipartite subgraphs in triangle-free graphs with maximum degree three, J. Graph Theory 10 (1986), 477–504.
F. Chung and R. Graham: On graphs not containing prescribed induced subgraphs, in: A tribute to Paul Erdős, Cambridge Univ. Press, Cambridge, 1990, 111–120.
C. Edwards: Some extremal properties of bipartite subgraphs, Canad. J. Math. 25 (1973), 475–485.
P. Erdős: On even subgraphs of graphs, Mat. Lapok 18 (1967), 283–288.
P. Erdős: Problems and results in graph theory and combinatorial analysis, in: Proc. of the 5th British Combinatorial Conference (Univ. Aberdeen, 1975), Congressus Numerantium XV (1976), 169–192.
P. Erdős, R. Faudree, J. Pach and J. Spencer: How to make a graph bipartite, J. Combin. Theory Ser. B 45 (1988), 86–98.
P. Erdős, E. Győri 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. Keevash and B. Sudakov: Local density in graphs with forbidden subgraphs, Combinatorics, Probability and Computing 12 (2003), 139–153.
P. Keevash and B. Sudakov: Sparse halves in triangle-free graphs, J. Combin. Theory Ser. B 96(4) (2006), 614–620.
J. Komlós and M. Simonovits: Szemerédi’s Regularity Lemma and its applications in graph theory, in: Combinatorics, Paul Erdős is eighty, Vol. 2, János Bolyai Math. Soc., Budapest, 1996, 295–352.
M. Krivelevich: On the edge distribution in triangle-free graphs, J. Combin. Theory Ser. B 63 (1995), 245–260.
J. Shearer: A note on bipartite subgraphs of triangle-free graphs, Random Structures Algorithms 3 (1992), 223–226.
E. Szemerédi: Regular partitions of graphs, in: Proc. Colloque Inter. CNRS, 260, CNRS, Paris, 1978, 399–401.
P. Turán: Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941), 436–452.
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Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.
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Sudakov, B. Note making a K 4-free graph bipartite. Combinatorica 27, 509–518 (2007). https://doi.org/10.1007/s00493-007-2238-0
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DOI: https://doi.org/10.1007/s00493-007-2238-0