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Decomposing Graphs into Edges and Triangles

Part of: Graph theory

Published online by Cambridge University Press:  13 March 2019

DANIEL KRÁL' ,
BERNARD LIDICKÝ ,
TAÍSA L. MARTINS  and
YANITSA PEHOVA
DANIEL KRÁL'
Affiliation:
Mathematics Institute, DIMAP and Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK (e-mail: [email protected])
BERNARD LIDICKÝ
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA 50011, USA (e-mail: [email protected])
TAÍSA L. MARTINS
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (e-mail: [email protected], [email protected])
YANITSA PEHOVA
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (e-mail: [email protected], [email protected])
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Abstract

We prove the following 30 year-old conjecture of Győri and Tuza: the edges of every n-vertex graph G can be decomposed into complete graphs C1,. . .,Cℓ of orders two and three such that |C1|+···+|Cℓ| ≤ (1/2+o(1))n2. This result implies the asymptotic version of the old result of Erdős, Goodman and Pósa that asserts the existence of such a decomposition with ℓ ≤ n2/4.

MSC classification

Primary: 05C70: Factorization, matching, partitioning, covering and packing
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 3 , May 2019 , pp. 465 - 472
Copyright
Copyright © Cambridge University Press 2019 

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Footnotes

This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement 648509). This publication reflects only its authors' view; the European Research Council Executive Agency is not responsible for any use that may be made of the information it contains.

§

The first author was also supported by the Engineering and Physical Sciences Research Council (EPSRC) Standard Grant EP/M025365/1.

This author was supported in part by NSF grant DMS-1600390.

∥

This author was also supported by the CNPq Science Without Borders grant 200932/2014-4.

References

[1] Baber, R. and Talbot, J. (2011) Hypergraphs do jump. Combin. Probab. Comput. 20 161–171.Google Scholar
[2] Baber, R. and Talbot, J. (2014) A solution to the 2/3 conjecture. SIAM J. Discrete Math. 28 756–766.Google Scholar
[3] Balogh, J., Hu, P., Lidický, B. and Liu, H. (2014) Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. European J. Combin. 35 75–85.Google Scholar
[4] Balogh, J., Hu, P., Lidický, B. and Pfender, F. (2016) Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle. European J. Combin. 52 47–58.Google Scholar
[5] Balogh, J., Hu, P., Lidický, B., Pfender, F., Volec, J. and Young, M. (2017) Rainbow triangles in three-colored graphs. J. Combin. Theory Ser. B 126 83–113.Google Scholar
[6] Balogh, J., Hu, P., Lidický, B., Pikhurko, O., Udvari, B. and Volec, J. (2015) Minimum number of monotone subsequences of length 4 in permutations. Combin. Probab. Comput. 24 658–679.Google Scholar
[7] Chung, F. R. K. (1981) On the decomposition of graphs. SIAM J. Algebraic Discrete Methods 2 1–12.Google Scholar
[8] Coregliano, L. N. and Razborov, A. A. (2017) On the density of transitive tournaments. J. Graph Theory 85 12–21.Google Scholar
[9] Cummings, J., Král', D., Pfender, F., Sperfeld, K., Treglown, A. and Young, M. (2013) Monochromatic triangles in three-coloured graphs. J. Combin. Theory Ser. B 103 489–503.Google Scholar
[10] Das, S., Huang, H., Ma, J., Naves, H. and Sudakov, B. (2013) A problem of Erdős on the minimum number of k-cliques. J. Combin. Theory Ser. B 103 344–373.Google Scholar
[11] Erdős, P., Goodman, A. W. and Pósa, L. (1966) The representation of a graph by set intersections. Canad. J. Math. 18 106–112.Google Scholar
[12] Even-Zohar, C. and Linial, N. (2015) A note on the inducibility of 4-vertex graphs. Graphs Combin. 31 1367–1380.Google Scholar
[13] Falgas-Ravry, V., Marchant, E., Pikhurko, O. and Vaughan, E. R. (2015) The codegree threshold for 3-graphs with independent neighborhoods. SIAM J. Discrete Math. 29 1504–1539.Google Scholar
[14] Gethner, E., Hogben, L., Lidický, B., Pfender, F., Ruiz, A. and Young, M. (2017) On crossing numbers of complete tripartite and balanced complete multipartite graphs. J. Graph Theory 84 552–565.Google Scholar
[15] Glebov, R., Král', D. and Volec, J. (2016) A problem of Erdős and Sós on 3-graphs. Israel J. Math. 211 349–366.Google Scholar
[16] Goaoc, X., Hubard, A., de Joannis de Verclos, R., Sereni, J.-S. and Volec, J. (2015) Limits of order types. In 31st International Symposium on Computational Geometry, Vol. 34 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl–Leibniz-Zentrum für Informatik, pp. 300–314.Google Scholar
[17] Győri, E. and Keszegh, B. (2015) On the number of edge-disjoint triangles in K 4-free graphs. arXiv:1506.03306Google Scholar
[18] Győri, E. and Keszegh, B. (2017) On the number of edge-disjoint triangles in K 4-free graphs. Electron. Notes Discrete Math. 61 557–560.Google Scholar
[19] Győri, E. and Kostochka, A. V. (1979) On a problem of G. O. H. Katona and T. Tarján. Acta Math. Acad. Sci. Hungar. 34 321–327.Google Scholar
[20] Győri, E. and Tuza, Z. (1987) Decompositions of graphs into complete subgraphs of given order. Studia Sci. Math. Hungar. 22 315–320.Google Scholar
[21] Hatami, H., Hladký, J., Král', D., Norine, S. and Razborov, A. (2012) Non-three-colourable common graphs exist. Combin. Probab. Comput. 21 734–742.Google Scholar
[22] Haxell, P. E. and Rödl, V. (2001) Integer and fractional packings in dense graphs. Combinatorica 21 13–38.Google Scholar
[23] Hladký, J., Král', D. and Norin, S. (2017) Counting flags in triangle-free digraphs. Combinatorica 37 49–76.Google Scholar
[24] Kahn, J. (1981) Proof of a conjecture of Katona and Tarján. Period. Math. Hungar. 12 81–82.Google Scholar
[25] Kim, J., Kühn, D., Osthus, D. and Tyomkyn, M. (2016) A blow-up lemma for approximate decompositions. arXiv:1604.07282. https://doi.org/10.1090/tran/7411Google Scholar
[26] Král', D., Lidický, B., Martins, T. L. and Pehova, Y. (2017) Decomposing graphs into edges and triangles. arXiv 1710.08486v2Google Scholar
[27] Král', D., Liu, C.-H., Sereni, J.-S., Whalen, P. and Yilma, Z. B. (2013) A new bound for the 2/3 conjecture. Combin. Probab. Comput. 22 384–393.Google Scholar
[28] Král', D., Mach, L. and Sereni, J.-S. (2012) A new lower bound based on Gromov's method of selecting heavily covered points. Discrete Comput. Geom. 48 487–498.Google Scholar
[29] Král', D. and Pikhurko, O. (2013) Quasirandom permutations are characterized by 4-point densities. Geom. Funct. Anal. 23 570–579.Google Scholar
[30] Lidický, B. and Pfender, F. (2017) Semidefinite programming and Ramsey numbers. arXiv:1704.03592Google Scholar
[31] McGuinness, S. (1994) Greedy maximum-clique decompositions. Combinatorica 14 335–343.Google Scholar
[32] McGuinness, S. (1994) The greedy clique decomposition of a graph. J. Graph Theory 18 427–430.Google Scholar
[33] Razborov, A. A. (2007) Flag algebras. J. Symbolic Logic 72 1239–1282.Google Scholar
[34] Tuza, Z. (2001) Unsolved Combinatorial Problems, Part I. BRICS Lecture Series LS-01-1.Google Scholar
[35] Yuster, R. (2005) Integer and fractional packing of families of graphs. Random Struct. Alg. 26 110–118.Google Scholar