2009/08/31 by Christofides, Demetres, Kühn, Daniela, Osthus, Deryk
#05C35 #05C45 #05C70 #05D40 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.0908.4572
In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every α> 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + α)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 δmust have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every α> 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least (1/2 + α)n can be almost decomposed into edge-disjoint Hamilton cycles.