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Getting a Directed Hamilton Cycle Two Times Faster

  • CHOONGBUM LEE (a1), BENNY SUDAKOV (a1) and DAN VILENCHIK (a1)
Abstract

Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if, at time t, instead of a random direction one is allowed to choose the orientation of et, then whether or not it is possible to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p. the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.

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[1]Alon, N. and Spencer, J. (2000) The Probabilistic Method, second edition, Wiley.
[2]Azar, Y., Broder, A., Karlin, A. and Upfal, E. (1999) Balanced allocations. SIAM J. Comput. 29 180200.
[3]Balogh, J., Bollobás, B., Krivelevich, M., Muller, T. and Walters, M. (2011) Hamilton cycles in random geometric graphs. Ann. Appl. Probab. 21 10531072.
[4]Bohman, T. and Frieze, A. (2001) Avoiding a giant component. Random Struct. Alg. 19 7585.
[5]Bohman, T.Frieze, A. and Wormald, N. (2004) Avoidance of a giant component in half the edge set of a random graph. Random Struct. Alg. 25 432449.
[6]Bohman, T. and Kravitz, D. (2006) Creating a giant component. Combin. Probab. Comput. 15 489511.
[7]Bollobás, B. (1984) The evolution of sparse graphs. In Graph Theory and Combinatorics: Proc. Cambridge Combinatorial Conference in Honour of Paul Erdős, 1984 (Bollobás, B., ed.), pp. 335357.
[8]Bollobás, B., Fenner, T. and Frieze, A. (1985) An algorithm for finding Hamilton cycles in random graphs. In Proc. 17th Annual ACM Symposium on Theory of Computing, 1985, pp. 430439.
[9]Cooper, C. and Frieze, A. (1994) Hamilton cycles in a class of random directed graphs. J. Combin. Theory Ser. B 62 151163.
[10]Cooper, C. and Frieze, A. (2000) Hamilton cycles in random graphs and directed graphs. Random Struct. Alg. 16 369401.
[11]Diestel, R. (2005) Graph Theory, Vol. 173 of Graduate Texts in Mathematics, third edition, Springer.
[12]Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5A 1761.
[13]Erdős, P. and Turán, P. (1965) On some problems of a statistical group-theory I. Z. Wahrsch. Verw. Gebiete 4 175186.
[14]Flaxman, A., Gamarnik, D. and Sorkin, G. (2005) Embracing the giant component. Random Struct. Alg. 27 277289.
[15]Frieze, A. (1988) An algorithm for finding Hamilton cycles in random directed graphs. J. Algorithms 9 181204.
[16]Frieze, A. Personal communication.
[17]Komlós, J. and Szemerédi, E. (1983) Limit distribution for the existence of Hamilton cycles in random graphs. Discrete Math. 43 5563.
[18]Korshunov, A. (1976) Solution of a problem of Erdős and Rényi on Hamilton cycles in non-oriented graphs. Soviet Math. Dokl. 17 760764.
[19]Krivelevich, M., Loh, P. and Sudakov, B. (2009) Avoiding small subgraphs in Achlioptas processes. Random Struct. Alg. 34 165195.
[20]Krivelevich, M., Lubetzky, E. and Sudakov, B. (2010) Hamiltonicity thresholds in Achlioptas processes. Random Struct. Alg. 37 124.
[21]Pósa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359364.
[22]Robinson, R. and Wormald, N. C. (1994) Almost all regular graphs are Hamiltonian. Random Struct. Alg. 5 363374.
[23]Sinclair, A. and Vilenchik, D. (2010) Delaying satisfiability for random 2SAT. In APPROX-RANDOM, pp. 710723.
[24]Spencer, J. and Wormald, N. (2007) Birth control for giants. Combinatorica 27 587628.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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