Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

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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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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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 sparse random graphs of minimum degree at leastk

Journal of Graph Theory ◽

10.1002/(sici)1097-0118(200005)34:1<42::aid-jgt5>3.0.co;2-h ◽

2000 ◽

Vol 34 (1) ◽

pp. 42-59 ◽

Cited By ~ 8

Author(s):

B. Bollob�s ◽

C. Cooper ◽

T. I. Fenner ◽

A. M. Frieze

Keyword(s):

Random Graphs ◽

Minimum Degree ◽

Hamilton Cycles ◽

Edge Disjoint

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Edge-Disjoint Hamilton Cycles in Regular Graphs of Large Degree

Journal of the London Mathematical Society ◽

10.1112/jlms/s2-19.1.13 ◽

1979 ◽

Vol s2-19 (1) ◽

pp. 13-16 ◽

Cited By ~ 14

Author(s):

Bill Jackson

Keyword(s):

Large Degree ◽

Regular Graphs ◽

Hamilton Cycles ◽

Edge Disjoint

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Edge-Disjoint Induced Subgraphs with Given Minimum Degree

The Electronic Journal of Combinatorics ◽

10.37236/2882 ◽

2013 ◽

Vol 20 (1) ◽

Author(s):

Raphael Yuster

Keyword(s):

Positive Integer ◽

Minimum Degree ◽

Independent Set ◽

Induced Subgraphs ◽

Asymptotically Optimal ◽

Edge Disjoint

Let $h$ be a given positive integer. For a graph with $n$ vertices and $m$ edges, what is the maximum number of pairwise edge-disjoint {\em induced} subgraphs, each having minimum degree at least $h$? There are examples for which this number is $O(m^2/n^2)$. We prove that this bound is achievable for all graphs with polynomially many edges. For all $\epsilon > 0$, if $m \ge n^{1+\epsilon}$, then there are always $\Omega(m^2/n^2)$ pairwise edge-disjoint induced subgraphs, each having minimum degree at least $h$. Furthermore, any two subgraphs intersect in an independent set of size at most $1+ O(n^3/m^2)$, which is shown to be asymptotically optimal.

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An approximate version of Jackson’s conjecture

Combinatorics Probability Computing ◽

10.1017/s0963548320000152 ◽

2020 ◽

Vol 29 (6) ◽

pp. 886-899

Author(s):

Anita Liebenau ◽

Yanitsa Pehova

Keyword(s):

Bipartite Graph ◽

Small Proportion ◽

Complete Bipartite Graph ◽

Hamilton Cycle ◽

Hamilton Cycles ◽

Edge Disjoint ◽

Bipartite Digraph ◽

Bipartite Tournament ◽

Approximate Version

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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On Komlós’ tiling theorem in random graphs

Combinatorics Probability Computing ◽

10.1017/s0963548319000129 ◽

2019 ◽

Vol 29 (1) ◽

pp. 113-127

Author(s):

Rajko Nenadov ◽

Nemanja Škorić

Keyword(s):

Chromatic Number ◽

Random Graph ◽

Random Graphs ◽

Upper Bound ◽

Minimum Degree ◽

Arbitrary Graph ◽

Spanning Subgraph ◽

Delta G ◽

Image Position ◽

Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

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Optimal Construction of Edge-Disjoint Paths in Random Regular Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548300004284 ◽

2000 ◽

Vol 9 (3) ◽

pp. 241-263 ◽

Cited By ~ 9

Author(s):

ALAN M. FRIEZE ◽

LEI ZHAO

Keyword(s):

Polynomial Time ◽

Regular Graph ◽

Randomized Algorithm ◽

Constant Factor ◽

Disjoint Paths ◽

Regular Graphs ◽

Edge Disjoint ◽

Random Regular Graph ◽

Arbitrary Graphs ◽

Optimal Construction

Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is [Nscr ][Pscr ]-complete, although it is in [Pscr ] if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.

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