Hamilton cycles in dense vertex-transitive graphs

A famous conjecture of Lovasz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.

[1]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[2]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[3]  W. T. Gowers,et al.  Quasirandom Groups , 2007, Combinatorics, Probability and Computing.

[4]  László Babai.  Long cycles in vertex-transitive graphs , 1979, J. Graph Theory.

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

[6]  Tomasz Łuczak,et al.  R(Cn,Cn,Cn)≤(4+o(1))n , 1999 .

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

[8]  Vojtech Rödl,et al.  Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels , 2011, J. Comb. Theory, Ser. A.

[9]  Igor Pak,et al.  Hamiltonian paths in Cayley graphs , 2009, Discret. Math..

[10]  WERNER BALLMANN.  AUTOMORPHISM GROUPS , 2011 .

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

[12]  Joseph A. Gallian,et al.  Hamiltonian cycles and paths in Cayley graphs and digraphs - A survey , 1996, Discret. Math..

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

[14]  Dave Witte Morris,et al.  A survey: Hamiltonian cycles in Cayley graphs , 1984, Discret. Math..

[15]  János Komlós,et al.  The Regularity Lemma and Its Applications in Graph Theory , 2000, Theoretical Aspects of Computer Science.

[16]  H. Fleischner.  The square of every two-connected graph is Hamiltonian , 1974 .

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

[18]  William J. Cook,et al.  Combinatorial optimization , 1997 .

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

[20]  R. Gorenflo,et al.  Multi-index Mittag-Leffler Functions , 2014 .

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

[22]  Daniela Kühn,et al.  Packings in Dense Regular Graphs , 2005, Comb. Probab. Comput..

[23]  Dragan Marusic,et al.  Hamilton cycles and paths in vertex-transitive graphs - Current directions , 2009, Discret. Math..

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

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

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

[27]  A. Schrijver.  A Course in Combinatorial Optimization , 1990 .

[28]  Fan Chung Graham,et al.  Quasi-random graphs , 1988, Comb..

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

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

[31]  L. Babai.  Automorphism groups, isomorphism, reconstruction , 1996 .