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Tilings in Randomly Perturbed Dense Graphs

Part of: Graph theory

Published online by Cambridge University Press:  16 July 2018

JÓZSEF BALOGH ,
ANDREW TREGLOWN  and
ADAM ZSOLT WAGNER
JÓZSEF BALOGH
Affiliation:
Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA (e-mail: [email protected])
ANDREW TREGLOWN
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (e-mail: [email protected])
ADAM ZSOLT WAGNER
Affiliation:
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA (e-mail: [email protected])
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Abstract

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 2 , March 2019 , pp. 159 - 176
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Research partially supported by NSF grant DMS-1500121 and an Arnold O. Beckman Research Award (UIUC Campus Research Board 15006).

‡

Research supported by EPSRC grant EP/M016641/1.

References

[1] Alon, N. and Yuster, R. (1993) Threshold functions for H-factors. Combin. Probab. Comput. 2 137–144.Google Scholar
[2] Balogh, J., McDowell, A., Molla, T., and Mycroft, R. (2018). Triangle-tilings in graphs without large independent sets. Combin. Probab. Comput, 27 (4) 449–474. doi:10.1017/S0963548318000196Google Scholar
[3] Bedenknecht, W., Han, J., Kohayakawa, Y. and Mota, G. O. Powers of tight Hamilton cycles in randomly perturbed hypergraphs. Submitted.Google Scholar
[4] Bennett, P., Dudek, A. and Frieze, A. Adding random edges to create the square of a Hamilton cycle. Submitted.Google Scholar
[5] Bohman, T., Frieze, A., Krivelevich, M. and Martin, R. (2004) Adding random edges to dense graphs. Random Struct. Alg. 24 105–117.Google Scholar
[6] Bohman, T., Frieze, A. and Martin, R. (2003) How many edges make a dense graph Hamiltonian? Random Struct. Alg. 22 33–42.Google Scholar
[7] Böttcher, J., Han, J., Kohayakawa, Y., Montgomery, R., Parczyk, O. and Person, Y. Universality for bounded degree spanning trees in randomly perturbed graphs. Submitted.Google Scholar
[8] Böttcher, J., Montgomery, R., Parczyk, O. and Person, Y. Embedding spanning bounded degree graphs in randomly perturbed graphs. Submitted.Google Scholar
[9] Böttcher, J., Schacht, M. and Taraz, A. (2009) Proof of the bandwidth conjecture of Bollobás and Komlós. Math. Ann. 343 175–205.Google Scholar
[10] Dirac, G. A. (1952) Some theorems on abstract graphs. Proc. London Math. Soc. 2 69–81.Google Scholar
[11] Erdős, P. and Rényi, A. (1966) On the existence of a factor of degree one of a connected random graph. Acta Math. Acad. Sci. Hungar. 17 359–368.Google Scholar
[12] Gerke, S. and McDowell, A. (2015) Nonvertex-balanced factors in random graphs. J. Graph Theory 78 269–286.Google Scholar
[13] Hajnal, A. and Szemerédi, E. (1970) Proof of a conjecture of Erdős. In Combinatorial Theory and its Applications, II, János Bolyai Mathematical Society, pp. 601–623.Google Scholar
[14] Han, J. and Zhao, Y. Hamiltonicity in randomly perturbed hypergraphs. Submitted.Google Scholar
[15] Janson, S., Łuczak, T. and Ruciński, A., A. (2000) Random Graphs, Wiley.Google Scholar
[16] Johansson, A., Kahn, J. and Vu, V. (2008) Factors in random graphs. Random Struct. Alg. 33 1–28.Google Scholar
[17] Joos, F. and Kim, J. Spanning trees in randomly perturbed graphs. Submitted.Google Scholar
[18] Komlós, J. (2000) Tiling Turán theorems. Combinatorica 20 203–218.Google Scholar
[19] Komlós, J., Sárközy, G. and Szemerédi, E. (2001) Proof of the Alon–Yuster conjecture. Discrete Math. 235 255–269.Google Scholar
[20] Krivelevich, M., Kwan, M. and Sudakov, B. (2016) Cycles and matchings in randomly perturbed digraphs and hypergraphs. Combin. Probab. Comput. 25 909–927.Google Scholar
[21] Krivelevich, M., Kwan, M. and Sudakov, B. (2017) Bounded-degree spanning trees in randomly perturbed graphs. SIAM J. Discrete Math. 31 155–171.Google Scholar
[22] Krivelevich, M., Sudakov, B. and Tetali, P. (2006) On smoothed analysis in dense graphs and formulas. Random Struct. Alg. 29 180–193.Google Scholar
[23] Kühn, D. and Osthus, D. (2006) Critical chromatic number and the complexity of perfect packings in graphs. In SODA 2006: 17th ACM–SIAM Symposium on Discrete Algorithms, pp. 851–859.Google Scholar
[24] Kühn, D. and Osthus, D. (2009) The minimum degree threshold for perfect graph packings. Combinatorica 29 65–107.Google Scholar
[25] Łuczak, T. and Ruciński, A. (1991) Tree-matchings in graph processes. SIAM J. Discrete Math. 4 107–120.Google Scholar
[26] McDowell, A. and Mycroft, R. Hamilton ℓ-cycles in randomly perturbed hypergraphs. Submitted.Google Scholar
[27] Pósa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359–364.Google Scholar
[28] Ruciński, A. (1992) Matching and covering the vertices of a random graph by copies of a given graph. Discrete Math. 105 185–197.Google Scholar
[29] Szemerédi, E. (1978) Regular partitions of graphs. In Problémes Combinatoires et Théorie des Graphes, Vol. 260 of Colloques Internationaux CNRS, pp. 399–401.Google Scholar
[30] Treglown, A. (2007) The regularity lemma and applications to packings in graphs. MSc thesis, University of Birmingham.Google Scholar