Edge-disjoint Hamilton cycles in graphs

Christofides, Demetres; Kühn, Daniela; Osthus, Deryk

doi:10.1016/j.jctb.2011.10.005

Cited by 33 publications

(79 citation statements)

References 11 publications

(26 reference statements)

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“…A similar calculation to Claim 2 shows that E [N a,b ] = (1 ± 4ǫ) np 4 2 , and a similar concentration argument shows that N a,b is within ǫ · np 4 2 of its mean with probability 1 − o(n −3 ). Therefore, a union bound over all a, b completes the proof of this final claim, and the proof of Lemma 3.4 At this point, we could immediately apply the results of Section 3.1 to pack Γ with perfect matchings, which then correspond to Hamilton cycles in D. However, it is unfortunate that this would miss most of the edges of D, since not all edges of D are in correspondence with edges of Γ.…”

Section: Proof By Construction γ Is a Bipartite Graph With Partssupporting

confidence: 53%

“…From (a), there are (1 ± 5ǫ) np 4 2 such quadruples. The probability that a given quadruple is completely labeled by i is [(1 ± 1.03ǫ)κ] −4 by (b), so the expected codegree in Γ ′ i is (1 ± 10ǫ) n 2 p κ 4 .…”

Section: For Eachmentioning

confidence: 99%

“…Indeed, Nash-Williams discovered that the Dirac condition already guarantees not just one, but at least 5 224 n edge-disjoint Hamilton cycles. His questions in [22,23,24] started a line of investigation, leading to recent work by Christofides, Kühn, and Osthus [4], who answered one of his conjectures asymptotically by proving that minimum degree 1 2 + o(1) n is already enough to guarantee n 8 edge-disjoint Hamilton cycles. For random graphs, these "packings" with Hamilton cycles are even more complete.…”

Section: Introductionmentioning

confidence: 99%

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Packing Tight Hamilton Cycles in Uniform Hypergraphs

Bal¹,

Frieze²

2012

SIAM J. Discrete Math.

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Let H be a 3-uniform hypergraph with n vertices. A tight Hamilton cycle C ⊂ H is a collection of n edges for which there is an ordering of the vertices v 1 , . . . , v n such that every triple of consecutive vertices {v i , v i+1 , v i+2 } is an edge of C (indices are considered modulo n). We develop new techniques which enable us to prove that under certain natural pseudo-random conditions, almost all edges of H can be covered by edge-disjoint tight Hamilton cycles, for n divisible by 4. Consequently, we derive the corollary that random 3-uniform hypergraphs can be almost completely packed with tight Hamilton cycles whp, for n divisible by 4 and p not too small. Along the way, we develop a similar result for packing Hamilton cycles in pseudo-random digraphs with even numbers of vertices.

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Section: Proof By Construction γ Is a Bipartite Graph With Partssupporting

confidence: 53%

Section: For Eachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Packing Tight Hamilton Cycles in Uniform Hypergraphs

Bal¹,

Frieze²

2012

SIAM J. Discrete Math.

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“…We let

V^{'} \subseteq V false(G false)

be such that

false| V^{'} false| ⩽ 1

and

false| V (G) ∖ V^{'} false|

is even. The difference is that we now apply the following result of to each

G_{*}^{c} : = G^{c} false[V (G) ∖ V^{'} false]

to obtain the desired matchings

M_{i}^{c}

: for every

α > 0

, any sufficiently large

n

‐vertex graph with minimum degree

δ ⩾ (1 / 2 + α) n

contains at least

(δ - α n + \sqrt{n (2 δ - n)}) / 4

edge‐disjoint Hamilton cycles.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…The result in general cannot be improved: Indeed, for

δ > 1 / 2

the number of edges of the densest regular spanning subgraph of

G

is close to

false(δ + \sqrt{2 δ - 1} false) n^{2} / 4

(see ). So the bound in (iii) is asymptotically optimal, for example, if

n

is even and

scriptH

consists of Hamilton cycles.…”

Section: Introductionmentioning

confidence: 99%

A bandwidth theorem for approximate decompositions

Condon

Kim

Kühn

et al. 2018

Proc. London Math. Soc.

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We provide a degree condition on a regular n‐vertex graph G which ensures the existence of a near optimal packing of any family scriptH of bounded degree n‐vertex k‐chromatic separable graphs into G. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of Böttcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let δk be the infimum over all δ⩾1/2 ensuring an approximate Kk‐decomposition of any sufficiently large regular n‐vertex graph G of degree at least δn. Now suppose that G is an n‐vertex graph which is close to r‐regular for some r⩾(δk+ofalse(1false))n and suppose that H1,⋯,Ht is a sequence of bounded degree n‐vertex k‐chromatic separable graphs with 0true∑ie(Hi)⩽(1−ofalse(1false))e(G). We show that there is an edge‐disjoint packing of H1,⋯,Ht into G. If the Hi are bipartite, then r⩾(1/2+o(1))n is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs G of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.

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