Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-14T09:30:23.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs

Part of: Graph theory

Published online by Cambridge University Press:  14 March 2016

MICHAEL KRIVELEVICH ,
MATTHEW KWAN  and
BENNY SUDAKOV
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 6997801, Israel (e-mail: [email protected])
MATTHEW KWAN
Affiliation:
Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland (e-mail: [email protected], [email protected])
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 6 , November 2016 , pp. 909 - 927
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bohman, T., Frieze, A., Krivelevich, M. and Martin, R. (2004) Adding random edges to dense graphs. Random Struct. Alg. 24 105–117.Google Scholar
[2] Bohman, T., Frieze, A. and Martin, R. (2003) How many random edges make a dense graph Hamiltonian? Random Struct. Alg. 22 33–42.Google Scholar
[3] Bondy, J. (1975) Pancyclic graphs: Recent results. In Infinite and Finite Sets: To Paul Erdős on his 60th Birthday, Vol. 1 (Hajnal, A., Rado, R. and Sós, V. T., eds), Vol. 10 of Colloquia Mathematica Societatis János Bolyai, North-Holland, pp. 181–187.Google Scholar
[4] Dirac, G. A. (1952) Some theorems on abstract graphs. Proc. London Math. Soc. 3 69–81.Google Scholar
[5] Dudek, A. and Frieze, A. (2011) Loose Hamilton cycles in random uniform hypergraphs. Electron. J. Combin. 18 #48.Google Scholar
[6] Frieze, A. and Krivelevich, M. (2005) On packing Hamilton cycles in ϵ-regular graphs. J. Combin. Theory Ser. B 94 159–172.Google Scholar
[7] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Cambridge University Press.Google Scholar
[8] Johansson, A., Kahn, J. and Vu, V. (2008) Factors in random graphs. Random Struct. Alg. 33 1–28.Google Scholar
[9] Karp, R. M. (1972) Reducibility among combinatorial problems. In Complexity of Computer Computations (Miller, R. E. and Thatcher, J. W., eds), Plenum, pp. 85–104.Google Scholar
[10] Keevash, P., Kühn, D., Mycroft, R. and Osthus, D. (2011) Loose Hamilton cycles in hypergraphs. Discrete Math. 311 544–559.Google Scholar
[11] Komlós, J. and Simonovits, M. (1996) Szemerédi's Regularity Lemma and its applications in graph theory. In Combinatorics: Paul Erdős is Eighty (Miklós, D., Sós, V., and Szőnyi, T., eds.), Vol. 2 of Bolyai Society Mathematical Studies, János Bolyai Mathematical Society, pp. 295–352.Google Scholar
[12] Korshunov, A. D. (1976) Solution of a problem of Erdős and Rényi on Hamilton cycles in nonoriented graphs. Soviet Math. Doklady 17 760–764.Google Scholar
[13] Krivelevich, M., Sudakov, B. and Tetali, P. (2006) On smoothed analysis in dense graphs and formulas. Random Struct. Alg. 29 180–193.Google Scholar
[14] Kühn, D., Lapinskas, J., Osthus, D. and Patel, V. (2014) Proof of a conjecture of Thomassen on Hamilton cycles in highly connected tournaments. Proc. London Math. Soc. 109 733–762.CrossRefGoogle Scholar
[15] Łuczak, T., Ruciński, A. and Gruszka, J. (1996) On the evolution of a random tournament. Discrete Math. 148 311–316.Google Scholar
[16] Moon, J. (1968) Topics on Tournaments, Holt, Rinehart and Winston.Google Scholar
[17] Moon, J. and Moser, L. (1962) Almost all tournaments are irreducible. Canadian Math. Bulletin 5 61–65.Google Scholar
[18] Pokrovskiy, A. (2014) Edge disjoint Hamiltonian cycles in highly connected tournaments. Int. Math. Res. Not. To appear.Google Scholar
[19] Pósa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359–364.Google Scholar
[20] Rödl, V. and Ruciński, A. (2010) Dirac-type questions for hypergraphs: A survey (or more problems for Endre to solve). In An Irregular Mind: Szemerédi is 70 (Bárány, I. and Solymosi, J., eds), Vol. 21 of Bolyai Society Mathematical Studies, Springer and János Bolyai Mathematical Society, pp. 561–590.Google Scholar
[21] Spielman, D. A. and Teng, S.-H. (2003) Smoothed analysis: Motivation and discrete models. In Algorithms and Data Structures: 8th International Workshop, WADS 2003 (Dehne, F., Sack, J. R., and Smid, M., eds), Vol. 2748 of Lecture Notes in Computer Science, Springer, pp. 256–270.Google Scholar
[22] Spielman, D. A. and Teng, S.-H. (2004) Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. J. Assoc. Comput. Mach. 51 385–463.Google Scholar
[23] Tao, T. (2006) Szemerédi's Regularity Lemma revisited. Contrib. Discrete Math. 1 8–28.Google Scholar