Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree d = (1/2+a)n. For any constant a > 0, we give an optimal answer in the following sense: let reg_even(n,d) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree d. Then the number of edge-disjoint Hamilton cycles we find equals reg_even(n,d)/2. The value of reg_even(n,d) is known for infinitely many values of n and d. We also extend our results to graphs G of minimum degree d >= n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of K\"uhn and Osthus on Hamilton decomposition of robustly expanding regular graphs.

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

[2]  Ryan R. Martin,et al.  Relating minimum degree and the existence of a k-factor , 2010 .

[3]  Bill Jackson,et al.  Edge-Disjoint Hamilton Cycles in Regular Graphs of Large Degree , 1979 .

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

[5]  Daniela Kühn,et al.  Edge‐disjoint Hamilton cycles in random graphs , 2011, Random Struct. Algorithms.

[6]  Harold Davenport.  On the Minimum of a Ternary Cubic Form , 1944 .

[7]  P. Katerinis,et al.  Minimum degree of a graph and the existence ofk-factors , 1985 .

[8]  W. T. Tutte.  The Factors of Graphs , 1952, Canadian Journal of Mathematics.

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

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

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

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

[13]  Stephen G. Hartke,et al.  Random partitions and edge-disjoint Hamiltonian cycles , 2013, J. Comb. Theory, Ser. B.

[14]  Guojun Li.  Edge disjoint Hamilton cycles in graphs , 2000, J. Graph Theory.

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

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