Edge-disjoint Hamilton cycles in graphs

Details Edge-disjoint Hamilton cycles in graphs Edge-disjoint Hamilton cycles in graphs

2009-08-31

arxiv.org/abs/0908.4572v1

Mathematics

Combinatorics

26 pages, 3 figures

Scientific paper

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \alpha > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least $(1/2 + \alpha)n$ can be almost decomposed into edge-disjoint Hamilton cycles.

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