17,502 research outputs found
Proof of Koml\'os's conjecture on Hamiltonian subsets
Koml\'os conjectured in 1981 that among all graphs with minimum degree at
least , the complete graph minimises the number of Hamiltonian
subsets, where a subset of vertices is Hamiltonian if it contains a spanning
cycle. We prove this conjecture when is sufficiently large. In fact we
prove a stronger result: for large , any graph with average degree at
least contains almost twice as many Hamiltonian subsets as ,
unless is isomorphic to or a certain other graph which we
specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ
On disjoint matchings in cubic graphs
For and a cubic graph let denote the maximum number
of edges that can be covered by matchings. We show that and . Moreover, it turns out that
.Comment: 41 pages, 8 figures, minor chage
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