Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments

Abstract A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into ( n − 1 ) / 2 edge-disjoint Hamilton cycles. We prove this conjecture for large n . In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdős on packing Hamilton cycles in random tournaments. As corollaries to the main result, we also obtain several results on packing Hamilton cycles in undirected graphs, giving e.g. the best known result on a conjecture of Nash-Williams. We also apply our result to solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover and Punnen as well as Alon, Gutin and Krivelevich.

[1]  Bill Jackson,et al.  Long paths and cycles in oriented graphs , 1981, J. Graph Theory.

[2]  D. Kuhn,et al.  Surveys in Combinatorics 2009: Embedding large subgraphs into dense graphs , 2009, 0901.3541.

[3]  Roland Häggkvist,et al.  Hamilton Cycles in Oriented Graphs , 1993, Combinatorics, Probability and Computing.

[4]  Béla Bollobás,et al.  Oriented Hamilton Cycles in Oriented Graphs , 1997 .

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

[6]  Daniela Kühn,et al.  A survey on Hamilton cycles in directed graphs , 2010, Eur. J. Comb..

[7]  Timothy W. Tillson,et al.  A Hamiltonian Decomposition of K & , 2 m > 8 , 2003 .

[8]  J. Moon.  Topics on tournaments , 1968 .

[9]  János Komlós,et al.  An algorithmic version of the blow-up lemma , 1998, Random Struct. Algorithms.

[10]  Béla Bollobás,et al.  On matchings and hamiltonian cycles in random graphs Annals of Discrete Mathematics 28 , 1985 .

[11]  Timothy W. Tillson.  A Hamiltonian decomposition of K2m*, 2m >= 8 , 1980, J. Comb. Theory B.

[12]  Daniela Kühn,et al.  Hamiltonian degree sequences in digraphs , 2008, J. Comb. Theory, Ser. B.

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

[14]  C. Thomassen.  Edge-Disjoint Hamiltonian Paths and Cycles in Tournaments , 1982 .

[15]  Noga Alon,et al.  Testing subgraphs in directed graphs , 2003, STOC '03.

[16]  Deryk Osthus,et al.  An exact minimum degree condition for Hamilton cycles in oriented graphs , 2008, 0801.0394.

[17]  Endre Szemerédi,et al.  Proof of the Seymour conjecture for large graphs , 1998 .

[18]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .

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

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

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

[22]  Bruce A. Reed,et al.  Edge coloring regular graphs of high degree , 1997, Discret. Math..

[23]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

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

[25]  Daniela Kühn,et al.  A Semiexact Degree Condition for Hamilton Cycles in Digraphs , 2010, SIAM J. Discret. Math..

[26]  A. Srivastav,et al.  Algorithmic Chernoff-Hoeffding inequalities in integer programming , 1996 .

[27]  Noga Alon,et al.  Algorithms with large domination ratio , 2004, J. Algorithms.

[28]  Deryk Osthus,et al.  Hamilton decompositions of regular tournaments , 2009, 0908.3411.

[29]  Deryk Osthus,et al.  Approximate Hamilton Decompositions of Robustly Expanding Regular Digraphs , 2013, SIAM J. Discret. Math..

[30]  János Komlós,et al.  Blow-up Lemma , 1997, Combinatorics, Probability and Computing.

[31]  Carsten Thomassen,et al.  Surveys in Combinatorics: Long cycles in digraphs with constraints on the degrees , 1979 .

[32]  Abraham P. Punnen,et al.  The travelling salesman problem: new solvable cases and linkages with the development of approximation algorithms , 1997 .

[33]  D. Kuhn,et al.  Proof of the 1-factorization and Hamilton Decomposition Conjectures , 2014, 1401.4183.

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

[35]  Daniela Kühn,et al.  A Dirac-Type Result on Hamilton Cycles in Oriented Graphs , 2007, Combinatorics, Probability and Computing.

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

[37]  Jean-Claude Bermond,et al.  Cycles in digraphs- a survey , 1981, J. Graph Theory.

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

[39]  Gregory Gutin,et al.  TSP tour domination and Hamilton cycle decompositions of regular digraphs , 2001, Oper. Res. Lett..

[40]  Daniela Kühn,et al.  Finding Hamilton cycles in robustly expanding digraphs , 2012, J. Graph Algorithms Appl..

[41]  Vojtech Rödl,et al.  The Algorithmic Aspects of the Regularity Lemma , 1994, J. Algorithms.

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