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Packing Hamilton Cycles Online

Part of: Graph theory

Published online by Cambridge University Press:  22 March 2018

JOSEPH BRIGGS ,
ALAN FRIEZE ,
MICHAEL KRIVELEVICH ,
PO-SHEN LOH  and
BENNY SUDAKOV
JOSEPH BRIGGS
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: jbriggs@andrew.cmu.edu, alan@random.math.cmu.edu, ploh@cmu.edu)
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: jbriggs@andrew.cmu.edu, alan@random.math.cmu.edu, ploh@cmu.edu)
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel (e-mail: krivelev@post.tau.ac.il)
PO-SHEN LOH
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: jbriggs@andrew.cmu.edu, alan@random.math.cmu.edu, ploh@cmu.edu)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland (e-mail: benjamin.sudakov@math.ethz.ch)
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Abstract

It is known that w.h.p. the hitting time τ2σ for the random graph process to have minimum degree 2σ coincides with the hitting time for σ edge-disjoint Hamilton cycles [4, 9, 13]. In this paper we prove an online version of this property. We show that, for a fixed integer σ ⩾ 2, if random edges of Kn are presented one by one then w.h.p. it is possible to colour the edges online with σ colours so that at time τ2σ each colour class is Hamiltonian.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C15: Coloring of graphs and hypergraphs

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Special Issue 4: Special Issue: Oberwolfach Workshop , July 2018 , pp. 475 - 495
Copyright
Copyright © Cambridge University Press 2018 

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References

[1] Ajtai, M., Komlós, J. and Szemerédi, E. (1985) First occurrence of Hamilton cycles in random graphs. In Cycles in Graphs (Alspach, B. R. and Godsil, C. D., eds), Vol. 115 of North-Holland Mathematics Studies, North-Holland, pp. 173–178.Google Scholar
[2] Bohman, T., Frieze, A., Krivelevich, M., Loh, P. and Sudakov, B. (2011) Ramsey games with giants. Random Struct. Alg. 38 68–99.Google Scholar
[3] Bollobás, B. (1984) The evolution of sparse graphs. In Graph Theory and Combinatorics: Proceedings of the Cambridge Combinatorial Conference in Honour of Paul Erdős (Bollobás, B., ed.), pp. 35–57.Google Scholar
[4] Bollobás, B. and Frieze, A. M. (1985) On matchings and Hamiltonian cycles in random graphs. Ann. Discrete Math. 28 23–46.Google Scholar
[5] Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5A 17–61.Google Scholar
[6] Erdős, P. and Rényi, A. (1961) On the strength of connectedness of a random graph. Acta. Math. Acad. Sci. Hungar. 8 261–267.Google Scholar
[7] Frieze, A. M. and Karoński, M. (2016) Introduction to Random Graphs, Cambridge University Press.Google Scholar
[8] Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.Google Scholar
[9] Knox, F., Kühn, D. and Osthus, D. (2015) Edge-disjoint Hamilton cycles in random graphs. Random Struct. Alg. 46 397–445.Google Scholar
[10] Komlós, J. and Szemerédi, E. (1983) Limit distribution for the existence of Hamilton cycles in random graphs. Discrete Math. 43 55–63.Google Scholar
[11] Korshunov, A. (1976) Solution of a problem of Erdős and Rényi on Hamilton cycles in non-oriented graphs. Soviet Math. Dokl. 17 760–764.Google Scholar
[12] Krivelevich, M., Lubetzky, E. and Sudakov, B. (2010) Hamiltonicity thresholds in Achlioptas processes. Random Struct. Alg. 37 1–24.Google Scholar
[13] Krivelevich, M. and Samotij, W. (2012) Optimal packings of Hamilton cycles in sparse random graphs. SIAM J. Discrete Math. 26 964–982.Google Scholar
[14] Lee, C., Sudakov, B. and Vilenchik, D. (2012) Getting a directed Hamilton cycle two times faster. Combin. Probab. Comput. 21 773–801.Google Scholar
[15] Pósa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359–364.Google Scholar