Edge-disjoint Hamiltonian Cycles Research Articles

Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs

We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a densek-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

Disjoint Hamilton cycles in transposition graphs

Most network topologies that have been studied have been subgraphs of transposition graphs. Edge-disjoint Hamilton cycles are important in network topologies for improving fault-tolerance and distribution of messaging traffic over the network. Not much was known about edge-disjoint Hamilton cycles in general transposition graphs until recently Hung produced a construction of 4 edge-disjoint Hamilton cycles in the 5-dimensional transposition graph and showed how edge-disjoint Hamilton cycles in (n+1)-dimensional transposition graphs can be constructed inductively from edge-disjoint Hamilton cycles in n-dimensional transposition graphs. In the same work it was conjectured that n-dimensional transposition graphs have n−1 edge-disjoint Hamilton cycles for all n greater than or equal to 5. In this paper we provide an edge-labelling for transposition graphs and, by considering known Hamilton cycles in labelled star subgraphs of transposition graphs, are able to provide an extra edge-disjoint Hamilton cycle at the inductive step from dimension n to n+1, and thereby prove the conjecture.

Edge Disjoint Hamiltonian Cycles in Gaussian Networks

Gaussian networks are degree four symmetric networks and these are designed based on the concept of Gaussian integers. The Gaussian network can be described in terms of a generator $\alpha =a+bi$ , where $a$ and $b$ are integers and $i = \sqrt{-1}$ . When $\gcd (a,b)=1$ , how to find edge disjoint Hamiltonian cycles has been shown before. In this paper for any generator $\alpha =a+bi,$ even when $\gcd (a, b)=d > 1$ , how to obtain two edge disjoint Hamiltonian cycles in these networks is described.

Cycles and matchings in randomly perturbed digraphs and hypergraphs

We consider several situations where “typical” structures have certain spanning substructures (in particular, Hamilton cycles), but where worst-case extremal examples do not. In these situations we show that the extremal examples are “fragile” in that after a modest random perturbation our desired substructures will typically appear. This builds on a sizeable existing body of research. Our first theorem is that adding linearly many random edges to a dense k-uniform hypergraph typically ensures the existence of a perfect matching or a loose Hamilton cycle. We outline the proof of this theorem, which involves a nonstandard application of Szemerédi's regularity lemma to “beat the union bound”; this might be of independent interest. Our next theorem is that digraphs with certain strong expansion properties are pancyclic. This implies that adding a linear number of random edges typically makes a dense digraph pancyclic. Our final theorem is that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

Hamiltonian cycles in spanning subgraphs of line graphs

Let G be a graph and e=uv an edge in G (also a vertex in the line graph L(G) of G). Then e is in two cliques EG(u) and EG(v) with EG(u)∩EG(v)={e} of L(G), that correspond to all edges incident with u and v in G respectively. Let SL(G) be any spanning subgraph of L(G) such that every vertex e=uv is adjacent to at least min{dG(u)−1,⌈34dG(u)+12⌉} vertices of EG(u) and to at least min{dG(v)−1,⌈34dG(v)+12⌉} vertices of EG(v). Then if L(G) is Hamiltonian, we show that SL(G) is Hamiltonian. As a corollary we obtain a lower bound on the number of edge-disjoint Hamiltonian cycles in L(G).

Edge disjoint Hamiltonian cycles in Eisenstein–Jacobi networks

Many communication algorithms in parallel systems can be efficiently solved by obtaining edge disjoint Hamiltonian cycles in the interconnection topology of the network. The Eisenstein–Jacobi (EJ) network generated by α=a+bρ, where ρ=(1+i3)/2, is a degree six symmetric interconnection network. The hexagonal network is a special case of the EJ network that can be obtained by α=a+(a+1)ρ. Generating three edge disjoint Hamiltonian cycles in the EJ network with generator α=a+bρ for gcd(a,b)=1 has been shown before. However, this problem has not been solved when gcd(a,b)=d>1. In this paper, some results to this problem are given.

A Note on Edge-Disjoint Hamilton Cycles in Line Graphs

It is well known that if a graph G contains a spanning closed trail, then its line graph L(G) is Hamiltonian. In this note, it is proved that if a graph G with minimum degree at least 4k has k edge-disjoint spanning closed trails, then L(G) contains k edge-disjoint Hamilton cycles.

A note on Hamiltonian decomposition of Bubble-Sort graphs

The Bubble-Sort graph, denoted by (n is positive integer), is a special class of Cayley graph model. In 2009, Shi and Niu [Hamiltonian decomposition of some interconnection networks, in Combinatorial Optimization and Applications, D.-Z. Du, X. Hu, and P.M. Pardalos, eds., Springer, Huangshan, 2009, pp. 231–237.] proposed the following conjecture: (i) If n is odd then is the union of edge-disjoint Hamiltonian cycles. (ii) If n is even then is the union of edge-disjoint Hamiltonian cycles and a perfect matching. In this paper, we give a construction of the decomposition of Bubble-Sort graph with n odd using the decomposition of . Moreover, if the decomposition of is given using the decomposition of then the conjecture is proved.

Structural properties of subdivided-line graphs

Motivated by self-similar structures of Sierpiński graphs, we newly introduce the sub-divided-line graph operation Γ and define the n-iterated subdivided-line graph Γn(G) of a graph G. We then study structural properties of subdivided-line graphs such as edge-disjoint Hamilton cycles, hamiltonian-connectivity, hub sets, connected dominating sets, independent spanning trees, completely independent spanning trees, and book-embeddings which can be applied to problems on interconnection networks. From our results, the maximum number of edge-disjoint Hamilton cycles, the minimum cardinality of a hub set, the minimum cardinality of a connected dominating set, the maximum number of independent spanning trees and the maximum number of completely independent spanning trees in Sierpiński graphs, and upper bounds on the pagenumbers of Sierpiński graphs which are at most two times the optimums are obtained as corollaries. In particular, our results for edge-disjoint Hamilton cycles and hub sets on iterated subdivided-line graphs are generalizations of the previously known results on Sierpiński graphs, while our proofs are simpler than those for Sierpiński graphs.

A polynomial algorithm with approximation ratio 2/3 for the Asymmetric Maximum 2-Peripatetic Salesman Problem

A polynomial approximation algorithm with approximation ratio 2/3 and cubic running time is presented for the maximization version of the Asymmetric 2-Peripatetic Salesman Problem that consists in finding two edge-disjoint Hamiltonian cycles of maximum total weight in a complete weighted graph.

The property of edge-disjoint Hamiltonian cycles in transposition networks and hypercube-like networks

The presence of edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the network. Edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant hamiltonicity of an interconnection network. In this paper, we first study the property of edge-disjoint Hamiltonian cycles in transposition networks which form a subclass of Cayley graphs. The transposition networks include other famous network topologies as their subgraphs, such as meshes, hypercubes, star graphs, and bubble-sort graphs. We first give a novel decomposition of transposition networks. Using the proposed decomposition, we show that n-dimensional transposition network with n⩾5 contains four edge-disjoint Hamiltonian cycles. By using the similar approach, we present a linear time algorithm to construct two edge-disjoint Hamiltonian cycles on crossed cubes which is a variation of hypercubes. The proposed approach can be easily applied to construct two edge-disjoint Hamiltonian cycles on the other variations of hypercubes.

Optimal covers with Hamilton cycles in random graphs

A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has size \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szab\'o recently initiated research on the `dual' problem, where one asks for a set of Hamilton cycles covering all edges of G. Our main result states that for log^{117}n / n < p < 1-n^{-1/8}, a.a.s. the edges of G_n,p can be covered by \lceil Delta(G_n,p)/2 \rceil Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szab\'o, which holds for p > n^{-1+\eps}. Our proof is based on a result of Knox, K\"uhn and Osthus on packing Hamilton cycles in pseudorandom graphs.

Proof of a conjecture of Thomassen on Hamilton cycles in highly connected tournaments

A conjecture of Thomassen from 1982 states that, for every k , there is an f ( k ) so that every strongly f ( k ) -connected tournament contains k edge-disjoint Hamilton cycles. A classical theorem of Camion, that every strongly connected tournament contains a Hamilton cycle, implies that f ( 1 ) = 1 . So far, even the existence of f ( 2 ) was open. In this paper, we prove Thomassen's conjecture by showing that f ( k ) = O ( k 2 log 2 k ) . This is best possible up to the logarithmic factor. As a tool, we show that every strongly 10 4 k log k -connected tournament is k-linked (which improves a previous exponential bound). The proof of the latter is based on a fundamental result of Ajtai, Komlós and Szemerédi on asymptotically optimal sorting networks.

Cores of random graphs are born Hamiltonian

Let $(G_t)_{t \geq 0}$ be the random graph process ($G_0$ is edgeless and $G_t$ is obtained by adding a uniformly distributed new edge to $G_{t-1}$), and let $\tau_k$ denote the minimum time $t$ such that the $k$-core of $G_t$ (its unique maximal subgraph with minimum degree at least $k$) is nonempty. For any fixed $k\geq 3$ the $k$-core is known to emerge via a discontinuous phase transition, where at time $t=\tau_k$ its size jumps from 0 to linear in the number of vertices with high probability. It is believed that for any $k\geq 3$ the core is Hamiltonian upon creation w.h.p., and Bollob\'as, Cooper, Fenner and Frieze further conjectured that it in fact admits $\lfloor(k-1)/2\rfloor$ edge-disjoint Hamilton cycles. However, even the asymptotic threshold for Hamiltonicity of the $k$-core in $G(n,p)$ was unknown for any $k$. We show here that for any fixed $k\ge 15$ the $k$-core of $G_t$ is w.h.p. Hamiltonian for all $t \geq \tau_k$, i.e., immediately as the $k$-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed $k$ the $k$-core contains $\lfloor (k-3)/2\rfloor$ edge-disjoint Hamilton cycles w.h.p. for all $t\geq \tau_k$.

Generalized Hypercubes: Edge-Disjoint Hamiltonian Cycles and Gray Codes

Some new classes of Hamming metric Gray codes over Zpn , where p is a prime and n is an integer power of 2, are described; then, how these Gray codes can be used to generate the maximum number of edge-disjoint Hamiltonian cycles in an n-dimensional generalized hypercube (GHC), Qpn, is shown. For Qpn, the number of edge-disjoint Hamiltonian cycles generated using these methods is n(p -1)/2 which is the maximum possible since the degree of each node in Qpn is n(p - 1). In addition, for any integers p and n, p not necessarily a prime and n not necessarily a power of 2, how to generate the maximum number of edge-disjoint Hamiltonian cycles in Qnp is also described.

Improving the panconnectedness property of locally twisted cubes

The n-dimensional locally twisted cube LTQn is a promising alternative to the hypercube because of its great properties. Not only is LTQn n-connected, but also meshes, torus, and edge-disjoint Hamiltonian cycles can embed in it. Ma and Xu [Panconnectivity of locally twisted cubes, Appl. Math. Lett. 19 (2006), pp. 681–685] investigated the panconnectivity of LTQn for flexible routing. In this paper, we combine panconnectivity with Hamiltonian connectedness to define Hamiltonian r-panconnectedness: a graph G of m vertices, m≥3, is Hamiltonian r-panconnected if for any three distinct vertices x, y, and z of G there exists a Hamiltonian path P of G such that P(1)=x, P(l+1)=y, and P(m)=z for every r≤l≤m−1−r, where P(i) denotes the ith vertex of P for 1≤i≤m. Then, we show that LTQn is Hamiltonian n-panconnected for n≥5. This property admits the path embedding via an intermediate node at any prescribed position, and our result achieves an improvement over that of Ma and Xu.

Hamilton decompositions of regular expanders: Applications

In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n−1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following:(i)an undirected analogue of our result on robust outexpanders;(ii)best possible bounds on the size of an optimal packing of edge-disjoint Hamilton cycles in a graph of minimum degree δ for a large range of values for δ.(iii)a similar result for digraphs of given minimum semidegree;(iv)an approximate version of a conjecture of Nash-Williams on Hamilton decompositions of dense regular graphs;(v)a verification of the ‘very dense’ case of a conjecture of Frieze and Krivelevich on packing edge-disjoint Hamilton cycles in random graphs;(vi)a proof of a conjecture of Erdős on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament.

Random partitions and edge-disjoint Hamiltonian cycles

Nash-Williams [1] proved that every graph with n vertices and minimum degree n/2 has at least ⌊5n/224⌋ edge-disjoint Hamiltonian cycles. In [2], he raised the question of determining the maximum number of edge-disjoint Hamiltonian cycles, showing an upper bound of ⌊(n+4)/8⌋.Let α(δ,n)=(δ+2δn−n2)/2. Christofides, Kühn, and Osthus [3] proved that for every ϵ>0, every graph G on a sufficiently large number n of vertices and minimum degree δ⩾n/2+ϵn contains α(δ,n)/2−ϵn/4 edge-disjoint Hamiltonian cycles. Their proof uses Szemerédiʼs Regularity Lemma, and hence the “sufficiently large” requirement on n is a strong condition.In this paper we prove a similar result using methods that do not rely on the Regularity Lemma. In particular, we prove that every graph on n vertices with minimum degree δ⩾n/2+3n3/4ln(n) contains α(δ,n)/2−3n7/8(lnn)1/4/2 edge-disjoint Hamiltonian cycles. Our proof rests on a structural result that is of independent interest: let G be a graph on n vertices, where n=pq. Then there exists a partition of the vertices of G into q parts of size p such that every vertex v has at least deg(v)/q−min{deg(v),p}⋅ln(n) neighbors in each part.

Edge-disjoint Hamilton cycles in random graphs

We show that provided log50n/ni¾?pi¾?1-n-1/4log9n we can with high probability find a collection of i¾?i¾?G/2i¾? edge-disjoint Hamilton cycles in G~Gn,p, plus an additional edge-disjoint matching of size i¾?n/2i¾? if i¾?G is odd. This is clearly optimal and confirms, for the above range of p, a conjecture of Frieze and Krivelevich. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 397-445, 2015

Embedding Two Edge-Disjoint Hamiltonian Cycles into Parity Cubes

Edge-disjoint Hamiltonian cycles are helpful to implement efficient and fault tolerant algorithms that require a ring structure. The n-dimensional parity cube, PQn, is a variant of hypercube. The existence of two edge-disjoint Hamiltonian cycles in parity cube is unknown. In this paper, we prove that there exist two edge-disjoint Hamiltonian cycles in n-dimensional parity cube with n≥4.