Edge‐disjoint Hamilton cycles in random graphs

We show that provided log50n/n≤p≤1−n−1/4log9n we can with high probability find a collection of ⌊δ(G)/2⌋ edge‐disjoint Hamilton cycles in G∼Gn,p , plus an additional edge‐disjoint matching of size ⌊n/2⌋ if δ(G) is odd. This is clearly optimal and confirms, for the above range of p, a conjecture of Frieze and Krivelevich. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 397–445, 2015

[1]  J. Petersen.  Die Theorie der regulären graphs , 1891 .

[2]  W. T. Tutte.  The Factorization of Linear Graphs , 1947 .

[3]  The Factors of Graphs , 1952 .

[4]  C. Nash-Williams,et al.  Hamiltonian arcs and circuits , 1971 .

[5]  A. Frieze.  ON MATCHINGS AND HAMILTON CYCLES IN RANDOM GRAPHS , 1988 .

[6]  Jean-Claude Bermond,et al.  Decomposition into Cycles I: Hamilton Decompositions , 1990 .

[7]  J. A. Bondy,et al.  Basic graph theory: paths and circuits , 1996 .

[8]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[9]  Guojun Li,et al.  Edge disjoint Hamilton cycles in graphs , 2000 .

[10]  Nicholas C. Wormald,et al.  Random Matchings Which Induce Hamilton Cycles and Hamiltonian Decompositions of Random Regular Graphs , 2001, J. Comb. Theory, Ser. B.

[11]  Benny Sudakov,et al.  Sparse pseudo-random graphs are Hamiltonian , 2003, J. Graph Theory.

[12]  Daniela Kühn,et al.  Multicoloured Hamilton cycles and perfect matchings in pseudo-random graphs , 2004 .

[13]  Alan M. Frieze,et al.  On packing Hamilton cycles in ?-regular graphs , 2005, J. Comb. Theory, Ser. B.

[14]  B. Sudakov,et al.  Pseudo-random Graphs , 2005, math/0503745.

[15]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[16]  Daniela Kühn,et al.  Multicolored Hamilton Cycles and Perfect Matchings in Pseudorandom Graphs , 2006, SIAM J. Discret. Math..

[17]  Michael Krivelevich,et al.  On two Hamilton cycle problems in random graphs , 2008 .

[18]  Benny Sudakov,et al.  On the Resilience of Hamiltonicity and Optimal Packing of Hamilton Cycles in Random Graphs , 2011, SIAM J. Discret. Math..

[19]  Alan M. Frieze,et al.  Packing tight Hamilton cycles in 3-uniform hypergraphs , 2011, SODA '11.

[20]  Alan M. Frieze,et al.  Packing tight Hamilton cycles in 3-uniform hypergraphs , 2012 .

[21]  Daniela Kühn,et al.  Approximate Hamilton decompositions of random graphs , 2012, Random Struct. Algorithms.

[22]  Daniela Kühn,et al.  Edge-disjoint Hamilton cycles in graphs , 2012, J. Comb. Theory, Ser. B.

[23]  Wojciech Samotij,et al.  Optimal Packings of Hamilton Cycles in Sparse Random Graphs , 2011, SIAM J. Discret. Math..

[24]  Daniela Kühn,et al.  Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree , 2012, Combinatorics, Probability and Computing.

[25]  Brian Alspach,et al.  Paley graphs have Hamilton decompositions , 2012, Discret. Math..

[26]  Alan M. Frieze,et al.  Packing hamilton cycles in random and pseudo-random hypergraphs , 2012, Random Struct. Algorithms.

[27]  Daniela Kühn,et al.  Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments , 2012, ArXiv.

[28]  Daniela Kühn,et al.  Hamilton decompositions of regular expanders: Applications , 2012, J. Comb. Theory, Ser. B.

[29]  Daniela Kühn,et al.  Optimal covers with Hamilton cycles in random graphs , 2014, Comb..

[30]  Michael Krivelevich,et al.  On covering expander graphs by hamilton cycles , 2011, Random Struct. Algorithms.