Edge disjoint Hamilton cycles in graphs

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

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Packing Hamilton Cycles Online

Combinatorics Probability Computing ◽

10.1017/s0963548318000159 ◽

2018 ◽

Vol 27 (4) ◽

pp. 475-495

Author(s):

JOSEPH BRIGGS ◽

ALAN FRIEZE ◽

MICHAEL KRIVELEVICH ◽

PO-SHEN LOH ◽

BENNY SUDAKOV

Keyword(s):

Random Graph ◽

Minimum Degree ◽

Hitting Time ◽

Online Version ◽

Hamilton Cycles ◽

Fixed Integer ◽

Edge Disjoint ◽

Colour Class

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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Edge Disjoint Hamilton Cycles in Knödel Graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2148 ◽

2016 ◽

Vol Vol. 17 no. 3 (Graph Theory) ◽

Author(s):

Palanivel Subramania Nadar Paulraja ◽

S Sampath Kumar

Keyword(s):

Maximum Degree ◽

Hamilton Cycle ◽

Hamilton Cycles ◽

Cycle Decomposition ◽

Edge Disjoint ◽

International Audience ◽

Appl Math

International audience The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $ \lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.

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Edge-disjoint Hamilton cycles in 4-regular planar graphs

Aequationes Mathematicae ◽

10.1007/bf02190157 ◽

1981 ◽

Vol 22 (1) ◽

pp. 42-45 ◽

Cited By ~ 7

Author(s):

J. A. Bondy ◽

R. Häggkvist

Keyword(s):

Planar Graphs ◽

Hamilton Cycles ◽

Edge Disjoint

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Edge-disjoint Hamilton cycles in random graphs

Random Structures and Algorithms ◽

10.1002/rsa.20510 ◽

2013 ◽

Vol 46 (3) ◽

pp. 397-445 ◽

Cited By ~ 22

Author(s):

Fiachra Knox ◽

Daniela Kühn ◽

Deryk Osthus

Keyword(s):

Random Graphs ◽

Hamilton Cycles ◽

Edge Disjoint

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Optimal Bounds for Disjoint Hamilton Cycles in Star Graphs

International Journal of Foundations of Computer Science ◽

10.1142/s0129054118500090 ◽

2018 ◽

Vol 29 (03) ◽

pp. 377-389 ◽

Cited By ~ 1

Author(s):

Parisa Derakhshan ◽

Walter Hussak

Keyword(s):

Interconnection Network ◽

Multiple Edge ◽

Star Graph ◽

Hamilton Cycles ◽

Star Graphs ◽

Edge Disjoint ◽

Spanning Subgraph ◽

Optimal Bounds ◽

Hamilton Decompositions ◽

Hamilton Decomposition

In interconnection network topologies, the [Formula: see text]-dimensional star graph [Formula: see text] has [Formula: see text] vertices corresponding to permutations [Formula: see text] of [Formula: see text] symbols [Formula: see text] and edges which exchange the positions of the first symbol [Formula: see text] with any one of the other symbols. The star graph compares favorably with the familiar [Formula: see text]-cube on degree, diameter and a number of other parameters. A desirable property which has not been fully evaluated in star graphs is the presence of multiple edge-disjoint Hamilton cycles which are important for fault-tolerance. The only known method for producing multiple edge-disjoint Hamilton cycles in [Formula: see text] has been to label the edges in a certain way and then take images of a known base 2-labelled Hamilton cycle under different automorphisms that map labels consistently. However, optimal bounds for producing edge-disjoint Hamilton cycles in this way, and whether Hamilton decompositions can be produced, are not known for any [Formula: see text] other than for the case of [Formula: see text] which does provide a Hamilton decomposition. In this paper we show that, for all n, not more than [Formula: see text], where [Formula: see text] is Euler’s totient function, edge-disjoint Hamilton cycles can be produced by such automorphisms. Thus, for non-prime [Formula: see text], a Hamilton decomposition cannot be produced. We show that the [Formula: see text] upper bound can be achieved for all even [Formula: see text]. In particular, if [Formula: see text] is a power of 2, [Formula: see text] has a Hamilton decomposable spanning subgraph comprising more than half of the edges of [Formula: see text]. Our results produce a better than twofold improvement on the known bounds for any kind of edge-disjoint Hamilton cycles in [Formula: see text]-dimensional star graphs for general [Formula: see text].

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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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