Perfect matchings extend to two or more hamiltonian cycles in hypercubes

Kreweras conjectured that every perfect matching of a hypercube $Q_n$ for $n\ge 2$ can be extended to a hamiltonian cycle of $Q_n$. Fink confirmed the conjecture to be true. It is more general to ask whether every perfect matching of $Q_n$ for $n\ge 2$ can be extended to two or more hamiltonian cycles of $Q_n$. In this paper, we prove that every perfect matching of $Q_n$ for $n\ge 4$ can be extended to at least $2^{2^{n?4}}$ different hamiltonian cycles of $Q_n$.     

Perfect matchings extending on subcubes to Hamiltonian cycles of hypercubes

Recently, Fink [J. Fink, Perfect matchings extend to Hamilton cycles in hypercubes, J. Combin. Theory Ser. B 97 (2007) 1074-1076] affirmatively answered Kreweras’ conjecture asserting that every perfect matching of the hypercube extends to a Hamiltonian cycle. We strengthen this result in the following way. Given a partition of the hypercube into subcubes of nonzero dimensions, we show for every perfect matching of the hypercube that it extends on these subcubes to a Hamiltonian cycle if and only if it interconnects them.     

Fault-free Hamiltonian cycles passing through a prescribed linear forest in 3-ary n-cube with faulty edges

The k-ary n-cube Qkn （n ≥2 and k ≥3） is one of the most popular interconnection networks. In this paper, we consider the problem of a fault- free Hamiltonian cycle passing through a prescribed linear forest （i.e., pairwise vertex-disjoint paths） in the 3-ary n-cube Qn^3 with faulty edges. The following result is obtained. Let E0 （≠θ） be a linear forest and F （≠θ） be a set of faulty edges in Q3 such that E0∩ F = 0 and ｜E0｜ ＋｜F｜ ≤ 2n - 2. Then all edges of E0 lie on a Hamiltonian cycle in Qn^3- F, and the upper bound 2n - 2 is sharp.     

Bipanconnectivity of faulty hypercubes with minimum degree

In this paper, we consider the conditionally faulty hypercube Q_n with n ? 2 where each vertex of Q_n is incident with at least m fault-free edges, 2 ? m ? n − 1. We shall generalize the limitation m ? 2 in all previous results of edge-bipancyclicity. We also propose a new edge-fault-tolerant bipanconnectivity called k-edge-fault-tolerant bipanconnectivity. A bipartite graph is k-edge-fault-tolerant bipanconnected if G − F remains bipanconnected for any F ⊂ E(G) with ∣F∣ ? k. For every integer m, under the same hypothesis, we show that Q_n is (n − 2)-edge-fault-tolerant edge-bipancyclic and bipanconnected, and the results are optimal with respect to the number of edge faults tolerated. This not only improves some known results on edge-bipancyclicity and bipanconnectivity of hypercubes, but also simplifies the proof.     

Hamiltonian cycles in k-partite graphs

Chen et al determined the minimum degree threshold for which a balanced $k$-partite graph has a Hamiltonian cycle. We give an asymptotically tight minimum degree condition for Hamiltonian cycles in arbitrary $k$-partite graphs in that all parts have at most $n/2$ vertices (a necessary condition). To do this, we first prove a general result that both simplifies the process of checking whether a graph $G$ is a robust expander and gives useful structural information in the case when $G$ is not a robust expander. Then we use this result to prove that any $k$-partite graph satisfying the minimum degree condition is either a robust expander or else contains a Hamiltonian cycle directly.     

Hamiltonian cycles and paths with a prescribed set of edges in hypercubes and dense sets

This paper studies techniques of finding hamiltonian paths and cycles in hypercubes and dense sets of hypercubes. This problem is, in general, easily solvable but here the problem was modified by the requirement that a set of edges has to be used in such path or cycle. The main result of this paper says that for a given n, any sufficiently large hypercube contains a hamiltonian path or cycle with prescribed n edges just when the family of the edges satisfies certain natural necessary conditions. Analogous results are presented for dense sets. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Hamiltonian cycles through prescribed edges of 4-connected maximal planar graphs

In 1956, W.T. Tutte proved that every 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. It is shown that Sanders’ result is best possible by constructing 4-connected maximal planar graphs with three edges a large distance apart such that any hamiltonian cycle misses one of them. If the maximal planar graph is 5-connected then such a construction is impossible.     

Hamiltonian cycles in -circulant digraphs

Let D be the circulant digraph with n vertices and connection set {2,3,c}. (Assume D is loopless and has outdegree 3.) Work of S. C. Locke and D. Witte implies that if n is a multiple of 6, c{(n/2)+2,(n/2)+3}, and c is even, then D does not have a hamiltonian cycle. For all other cases, we construct a hamiltonian cycle in D.     

Hamiltonian paths with prescribed edges in hypercubes

Given a set P of at most 2n-4 prescribed edges (n?5) and vertices u and v whose mutual distance is odd, the n-dimensional hypercube Q_n contains a hamiltonian path between u and v passing through all edges of P iff the subgraph induced by P consists of pairwise vertex-disjoint paths, none of them having u or v as internal vertices or both of them as endvertices. This resolves a problem of Caha and Koubek who showed that for any n?3 there exist vertices u,v and 2n-3 edges of Q_n not contained in any hamiltonian path between u and v, but still satisfying the condition above. The proof of the main theorem is based on an inductive construction whose basis for n=5 was verified by a computer search. Classical results on hamiltonian edge-fault tolerance of hypercubes are obtained as a corollary.     

On certain Hamiltonian cycles in planar graphs

The problem is considered under which conditions a 4-connected planar or projective planar graph has a Hamiltonian cycle containing certain prescribed edges and missing certain forbidden edges. The results are applied to obtain novel lower bounds on the number of distinct Hamiltonian cycles that must be present in a 5-connected graph that is embedded into the plane or into the projective plane with face-width at least five. Especially, we show that every 5-connected plane or projective plane triangulation on n vertices with no non-contractible cyles of length less than five contains at least distinct Hamiltonian cycles. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 81–96, 1999     

Hamiltonian cycles and 2-dominating induced cycles in claw-free graphs

Let G = (V, E) be a connected graph. For a vertex subset , G[S] is the subgraph of G induced by S. A cycle C (a path, respectively) is said to be an induced cycle (path, respectively) if G[V(C)] = C (G[V(P)] = P, respectively). The distance between a vertex x and a subgraph H of G is denoted by , where d(x, y) is the distance between x and y. A subgraph H of G is called 2-dominating if d(x, H) ≤ 2 for all . An induced path P of G is said to be maximal if there is no induced path P′ satisfying and . In this paper, we assume that G is a connected claw-free graph satisfying the following condition: for every maximal induced path P of length p ≥ 2 with end vertices u, v it holds:

正圆有向图中的弧不相交的Hamilton路和圈

2012年,Bang-Jensen和Huang(J.Combin.Theory Ser.B.2012,102:701-714)证明了2-弧强的局部半完全有向图可以分解为两个弧不相交的强连通生成子图当且仅当D不是偶圈的二次幂,并提出了任意3-强的局部竞赛图中包含两个弧不相交的Hamilton圈的猜想.主要研究正圆有向图中的弧不相交的Hamilton路和Hamilton圈,并证明了任意3-弧强的正圆有向图中包含两个弧不相交的Hamilton圈和任意4-弧强的正圆有向图中包含一个Hamilton圈和两个Hamilton路,使得它们两两弧不相交.由于任意圆有向图一定是正圆有向图,所得结论可以推广到圆有向图中.又由于圆有向图是局部竞赛图的子图类,因此所得结论说明对局部竞赛图的子图类――圆有向图,Bang-Jensen和Huang的猜想成立.     

Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and ω₁(F)=s, where ω₁(F) is the number of components of order one in F. We denote by σ₃(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ₃ conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ₃(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if σ₃(G)≥n+κ(G)+2m−1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and σ₃(G)≥n+κ(G)+2, then G is hamiltonian-connected.     

A fast parallel algorithm for finding Hamiltonian cycles in dense graphs

Suppose that 0<η<1 is given. We call a graph, G, on n vertices an η-Chvátal graph if its degree sequence d₁≤d₂≤?≤d_n satisfies: for k<n/2, d_k≤min{k+ηn,n/2} implies d_n−k−ηn≥n−k. (Thus for η=0 we get the well-known Chvátal graphs.) An -algorithm is presented which accepts as input an η-Chvátal graph and produces a Hamiltonian cycle in G as an output. This is a significant improvement on the previous best -algorithm for the problem, which finds a Hamiltonian cycle only in Dirac graphs (δ(G)≥n/2 where δ(G) is the minimum degree in G).     

Hamiltonian cycles in n-factor-critical graphs

A graph G is said to be n-factor-critical if G−S has a 1-factor for any SV(G) with |S|=n. In this paper, we prove that if G is a 2-connected n-factor-critical graph of order p with , then G is hamiltonian with some exceptions. To extend this theorem, we define a (k,n)-factor-critical graph to be a graph G such that G−S has a k-factor for any SV(G) with |S|=n. We conjecture that if G is a 2-connected (k,n)-factor-critical graph of order p with , then G is hamiltonian with some exceptions. In this paper, we characterize all such graphs that satisfy the assumption, but are not 1-tough. Using this, we verify the conjecture for k2.     

Hamiltonian cycles in bipartite quadrangulations on the torus

In this article, we shall prove that every bipartite quadrangulation G on the torus admits a simple closed curve visiting each face and each vertex of G exactly once but crossing no edge. As an application, we conclude that the radial graph of any bipartite quadrangulation on the torus has a hamiltonian cycle. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:143‐151, 2012     

Finding large cycles in Hamiltonian graphs

We show how to find in Hamiltonian graphs a cycle of length n^{Ω(1/loglogn)}=exp(Ω(logn/loglogn)). This is a consequence of a more general result in which we show that if G has a maximum degree d and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in O(n³) time a cycle in G of length k^Ω(1/logd). From this we infer that if G has a cycle of length k, then one can find in O(n³) time a cycle of length k^{Ω(1/(log(n/k)+loglogn))}, which implies the result for Hamiltonian graphs. Our results improve, for some values of k and d, a recent result of Gabow (2004) [11] showing that if G has a cycle of length k, then one can find in polynomial time a cycle in G of length . We finally show that if G has fixed Euler genus g and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in polynomial time a cycle in G of length f(g)k^Ω(1), running in time O(n²) for planar graphs.     

A generalization of Tutte’s theorem on Hamiltonian cycles in planar graphs

In 1956, W.T. Tutte proved that a 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. We prove that a planar graph G has a cycle containing a given subset X of its vertex set and any two prescribed edges of the subgraph of G induced by X if |X|≥3 and if X is 4-connected in G. If X=V(G) then Sanders’ result follows.