Hamilton Cycles, Minimum Degree, and Bipartite Holes

It is known that w.h.p. the hitting time τ2σ for the random graph process to have minimum degree 2σ coincides with the hitting time for σ edge-disjoint Hamilton cycles [4, 9, 13]. In this paper we prove an online version of this property. We show that, for a fixed integer σ ⩾ 2, if random edges of Kn are presented one by one then w.h.p. it is possible to colour the edges online with σ colours so that at time τ2σ each colour class is Hamiltonian.

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Matchings and Hamilton cycles in hypergraphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3457 ◽

2005 ◽

Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽

Author(s):

Daniela Kühn ◽

Deryk Osthus

Keyword(s):

Bipartite Graph ◽

Perfect Matching ◽

Minimum Degree ◽

Hamilton Cycle ◽

Uniform Hypergraph ◽

Hamilton Cycles ◽

Uniform Hypergraphs ◽

International Audience

International audience It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac's theorem on Hamilton cycles for $3$-uniform hypergraphs: We say that a $3$-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. We prove that for every $\varepsilon > 0$ there is an $n_0$ such that every $3$-uniform hypergraph of order $n \geq n_0$ whose minimum degree is at least $n/4+ \varepsilon n$ contains a Hamilton cycle. Our bounds on the minimum degree are essentially best possible.

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An exact minimum degree condition for Hamilton cycles in oriented graphs

Journal of the London Mathematical Society ◽

10.1112/jlms/jdn065 ◽

2008 ◽

Vol 79 (1) ◽

pp. 144-166 ◽

Cited By ~ 20

Author(s):

Peter Keevash ◽

Daniela Kühn ◽

Deryk Osthus

Keyword(s):

Minimum Degree ◽

Hamilton Cycles ◽

Oriented Graphs ◽

Degree Condition

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Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

Combinatorics Probability Computing ◽

10.1017/s0963548312000569 ◽

2012 ◽

Vol 22 (3) ◽

pp. 394-416 ◽

Cited By ~ 9

Author(s):

DANIELA KÜHN ◽

JOHN LAPINSKAS ◽

DERYK OSTHUS

Keyword(s):

General Result ◽

Minimum Degree ◽

Regular Graphs ◽

Hamilton Cycles ◽

Edge Disjoint ◽

Large Graph ◽

Spanning Subgraph ◽

Hamilton Decomposition

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.

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