Finding Hamilton cycles in robustly expanding digraphs

We provide an NC algorithm for nding Hamilton cycles in directed graphs with a certain robust expansion property. This property captures several known criteria for the existence of Hamilton cycles in terms of the degree sequence and thus we provide algorithmic proofs of (i) an ‘oriented’ analogue of Dirac’s theorem and (ii) an approximate version (for directed graphs) of Chv atal’s theorem. Moreover, our main result is used as a tool

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

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

[3]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[4]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[5]  N. Alon,et al.  The Probabilistic Method, Second Edition , 2000 .

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

[7]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

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

[9]  Béla Csaba,et al.  On the Bollobás–Eldridge Conjecture for Bipartite Graphs , 2007, Combinatorics, Probability and Computing.

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

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

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

[13]  V. Chvátal On Hamilton's ideals , 1972 .

[14]  M. Meyniel Une condition suffisante d'existence d'un circuit hamiltonien dans un graphe oriente , 1973 .

[15]  Marek Karpinski,et al.  On the Parallel Complexity of Hamiltonian Cycle and Matching Problem on Dense Graphs , 1993, J. Algorithms.

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

[17]  M. Simonovits,et al.  Szemeredi''s Regularity Lemma and its applications in graph theory , 1995 .

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

[19]  W. T. Tutte On Hamiltonian Circuits , 1946 .

[20]  E. Szemerédi Regular Partitions of Graphs , 1975 .

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

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

[23]  S. Janson,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[24]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

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

[26]  G. Lev,et al.  Size bounds and parallel algorithms for networks , 1980 .

[27]  G. Dirac Some Theorems on Abstract Graphs , 1952 .

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

[29]  John Adrian Bondy,et al.  A method in graph theory , 1976, Discret. Math..

[30]  Gábor N. Sárközy A fast parallel algorithm for finding Hamiltonian cycles in dense graphs , 2009, Discret. Math..