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Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

Published online by Cambridge University Press:  20 December 2012

DANIELA KÜHN ,
JOHN LAPINSKAS  and
DERYK OSTHUS
DANIELA KÜHN
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (e-mail: d.kuhn@bham.ac.uk, jal129@bham.ac.uk, d.osthus@bham.ac.uk)
JOHN LAPINSKAS
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (e-mail: d.kuhn@bham.ac.uk, jal129@bham.ac.uk, d.osthus@bham.ac.uk)
DERYK OSTHUS
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (e-mail: d.kuhn@bham.ac.uk, jal129@bham.ac.uk, d.osthus@bham.ac.uk)
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Abstract

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree δ=(1/2+α)n. For any constant α>0, we give an optimal answer in the following sense: let regeven(n,δ) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. Then the number of edge-disjoint Hamilton cycles we find equals regeven(n,δ)/2. The value of regeven(n,δ) is known for infinitely many values of n and δ. We also extend our results to graphs G of minimum degree δ ≥ 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 Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.

Keywords

Primary 05C45Secondary 05C3505C70
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 3 , May 2013 , pp. 394 - 416
Copyright
Copyright © Cambridge University Press 2012

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