Edge‐disjoint Hamiltonian cycles in hypertournaments

We introduce a method for reducing k‐tournament problems, for k ≥ 3, to ordinary tournaments, that is, 2‐tournaments. It is applied to show that a k‐tournament on n ≥ k + 1 + 24d vertices (when k ≥ 4) or on n ≥ 30d + 2 vertices (when k = 3) has d edge‐disjoint Hamiltonian cycles if and only if it is d‐edge‐connected. Ironically, this is proved by ordinary tournament arguments although it only holds for k ≥ 3. We also characterizatize the pancyclic k‐tournaments, a problem posed by Gutin and Yeo.(Our characterization is slightly incomplete in that we prove it only for n large compared to k.). © 2005 Wiley Periodicals, Inc. J Graph Theory     

Monochromatic Hamiltonian t‐tight Berge‐cycles in hypergraphs

In any r‐uniform hypergraph for 2 ≤ t ≤ r we define an r‐uniform t‐tight Berge‐cycle of length ?, denoted by C_?^{(r, t)}, as a sequence of distinct vertices v₁, v₂, … , v_?, such that for each set (v_i, v_{i + 1}, … , v_{i + t ? 1}) of t consecutive vertices on the cycle, there is an edge E_i of that contains these t vertices and the edges E_i are all distinct for i, 1 ≤ i ≤ ?, where ? + j ≡ j. For t = 2 we get the classical Berge‐cycle and for t = r we get the so‐called tight cycle. In this note we formulate the following conjecture. For any fixed 2 ≤ c, t ≤ r satisfying c + t ≤ r + 1 and sufficiently large n, if we color the edges of K_n^(r), the complete r‐uniform hypergraph on n vertices, with c colors, then there is a monochromatic Hamiltonian t‐tight Berge‐cycle. We prove some partial results about this conjecture and we show that if true the conjecture is best possible. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 34–44, 2008     

Four‐cycles in graphs without a given even cycle

We prove that every bipartite C_2l‐free graph G contains a C₄‐free subgraph H with e(H) ≥ e(G)/(l – 1). The factor 1/(l – 1) is best possible. This implies that ex(n, C_2l) ≤ 2(l – 1)ex(n, {C₄, C_2l}), which settles a special case of a conjecture of Erd?s and Simonovits. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 147–156, 2005     

Edge‐disjoint Hamilton cycles in random graphs

We show that provided we can with high probability find a collection of edge‐disjoint Hamilton cycles in , plus an additional edge‐disjoint matching of size if 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     

Hamilton cycles in 3‐out

Let G_3‐out denote the random graph on vertex set [n] in which each vertex chooses three neighbors uniformly at random. Note that G_3‐out has minimum degree 3 and average degree 6. We prove that the probability that G_3‐out is Hamiltonian goes to 1 as n tends to infinity. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Measuring length of business cycles across countries using a new non‐stationary unit‐root cyclical approach

This article examines the length of the cycles in the gross domestic product (GDP) real per capita series of 15 countries by means of new statistical techniques based on unit root cycles. We propose tests for unit root cycles at each of the frequencies of the process. Using this approach, we are able to approximate the number of periods per cycle. The results show that the cycles have a periodicity of approximately six years when the disturbances are white noise. However, if we permit autocorrelation, they may also occur at smaller intervals of time. Copyright © 2006 John Wiley & Sons, Ltd.     

Packing complete multipartite graphs with 4‐cycles

In this paper we completely solve the problem of finding a maximum packing of any complete multipartite graph with edge‐disjoint 4‐cycles, and the minimum leaves are explicitly given. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 107–127, 2001     

Packing tight Hamilton cycles in 3‐uniform hypergraphs

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 v₁,…,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. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Generating all cycles, chordless cycles, and Hamiltonian cycles with the principle of exclusion

An efficient method to generate all edge sets XE of a graph G=(V,E), which are vertex-disjoint unions of cycles, is presented. It can be tweaked to generate (i) all cycles, (ii) all cycles of cardinality 5, (iii) all chordless cycles, (iv) all Hamiltonian cycles.     

Solution of a conjecture of Tewes and Volkmann regarding extendable cycles in in‐tournaments

A directed cycle C of a digraph D is extendable if there exists a directed cycle C′ in D that contains all vertices of C and an additional one. In 1989, Hendry defined a digraph D to be cycle extendable if it contains a directed cycle and every non‐Hamiltonian directed cycle of D is extendable. Furthermore, D is fully cycle extendable if it is cycle extendable and every vertex of D belongs to a directed cycle of length three. In 2001, Tewes and Volkmann extended these definitions in considering only directed cycles whose length exceed a certain bound 3≤k<n: a digraph D is k ‐extendable if every directed cycle of length t, where k≤t<n, is extendable. Moreover, D is called fully k ‐extendable if D is k ‐extendable and every vertex of D belongs to a directed cycle of length k. An in‐tournament is an oriented graph such that the in‐neighborhood of every vertex induces a tournament. This class of digraphs which generalizes the class of tournaments was introduced by Bang‐Jensen, Huang and Prisner in 1993. Tewes and Volkmann showed that every connected in‐tournament D of order n with minimum degree δ≥1 is ( ) ‐extendable. Furthermore, if D is a strongly connected in‐tournament of order n with minimum degree δ=2 or , then D is fully ( ) ‐extendable. In this article we shall see that if , every vertex of D belongs to a directed cycle of length , which means that D is fully ( ) ‐extendable. This confirms a conjecture of Tewes and Volkmann. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 82–92, 2010     

The 3‐D bifurcation and limit cycles in a food‐chain model

In this paper, by using a corollary to the center manifold theorem, we show that the 3‐D food‐chain model studied by many authors undergoes a 3‐D Hopf bifurcation, and then we obtain the existence of limit cycles for the 3‐D differential system. The methods used here can be extended to many other 3‐D differential equation models. Copyright © 2008 John Wiley & Sons, Ltd.     

Orthogonal double covers by super‐extendable cycles

An Orthogonal Double Cover (ODC) of the complete graph K_n by an almost‐hamiltonian cycle is a decomposition of 2K_n into cycles of length n?1 such that the intersection of any two of them is exactly one edge. We introduce a new class of such decompositions. If n is a prime, the special structure of such a decomposition allows to expand it to an ODC of K_n+1 by an almost‐hamiltonian cycle. This yields the existence of an ODC of K_p+1 by an almost‐hamiltonian cycle for primes p of order 3 mod 4 and its eventual existence for arbitrary primes p. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 283–293, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10011     

Some results about zero‐cycles on abelian and semi‐abelian varieties

In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration of Suslin's singular homology group, , such that the successive quotients are isomorphic to a certain Somekawa K‐group.     

Gallai–Ramsey numbers for cycles

For given graphs G and H and an integer k, the Gallai–Ramsey number is defined to be the minimum integer n such that, in any k coloring of the edges of K_n, there exists a subgraph isomorphic to either a rainbow coloring of G or a monochromatic coloring of H. In this work, we consider Gallai–Ramsey numbers for the case when G=K₃ and H is a cycle of a fixed length.     

Short k‐radius sequences, k‐difference sequences and universal cycles

An $n$‐ary $k$‐radius sequence is a finite sequence of elements taken from an alphabet of size $n$ in which any two distinct elements occur within distance $k$ of each other somewhere in the sequence. The study of constructing short $k$‐radius sequences was motivated by some problems occurring in large data transfer. Let $f_k\left(n\right)$ be the shortest length of any $n$‐ary $k$‐radius sequence. We show that the conjecture $f_k\left(n\right)=\frac{n^2}{2k}+O\left(n\right)$ by Bondy et al is true for $k\le 4$, and determine the exact values of $f_2\left(n\right)$ for new infinitely many $n$. Further, we investigate new sequences which we call $k$‐difference, as they are related to $k$‐radius sequences and seem to be interesting in themselves. Finally, we answer a question about the optimal length of packing and covering analogs of universal cycles proposed by D?bski et al.     

Squares of Hamiltonian cycles in 3‐uniform hypergraphs

We show that every 3‐uniform hypergraph H = (V,E) with |V(H)| = n and minimum pair degree at least (4/5 + o(1))n contains a squared Hamiltonian cycle. This may be regarded as a first step towards a hypergraph version of the Pósa‐Seymour conjecture.     

完全r部图Kr（t）的{=C3，C2k}-强制分解（k≥4）的渐近存在性

研究基于顶点集V=Ui=1^rVi（其中｜Vi｜=t，i=1，2，……，r）的完全r部图Kr（t）的3圈和2k圈{C3，C2k}-强制分解（k≥4）的存在性问题．通过构造并运用Kr（t）的两种分解法，证明了Kr（t）的〈C3，C2k}-强制分解（k≥4）的渐近存在性，即对于任意给定的正整数k≥4，存在常数r0（k）=5k＋2，使得当r≥r0（k）时，Kr（t）的{C3，C2k}-强制分解存在的必要条件也是充分的．     

On weights of induced paths and cycles in claw‐free and K1,r‐free graphs

Let G be a K_1,r ‐free graph (r ≥ 3) on n vertices. We prove that, for any induced path or induced cycle on k vertices in G (k ≥ 2r − 1 or k ≥ 2r, respectively), the degree sum of its vertices is at most (2r − 2)(n − α) where α is the independence number of G. As a corollary we obtain an upper bound on the length of a longest induced path and a longest induced cycle in a K_1,r ‐free graph. Stronger bounds are given in the special case of claw‐free graphs (i.e., r = 3). Sharpness examples are also presented. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 131–143, 2001     

Decomposition of complete uniform multi‐hypergraphs into Berge paths and cycles

In 2015, Bryant, Horsley, Maenhaut, and Smith, generalizing a well‐known conjecture by Alspach, obtained the necessary and sufficient conditions for the decomposition of the complete multigraph into cycles of arbitrary lengths, where I is empty, when is even and I is a perfect matching, when is odd. Moreover, Bryant in 2010, verifying a conjecture by Tarsi, proved that the obvious necessary conditions for packing pairwise edge‐disjoint paths of arbitrary lengths in are also sufficient. In this article, first, we obtain the necessary and sufficient conditions for packing edge‐disjoint cycles of arbitrary lengths in . Then, applying this result, we investigate the analogous problem of the decomposition of the complete uniform multihypergraph into Berge cycles and paths of arbitrary given lengths. In particular, we show that for every integer , and , can be decomposed into Berge cycles and paths of arbitrary lengths, provided that the obvious necessary conditions hold, thereby generalizing a result by Kühn and Osthus on the decomposition of into Hamilton Berge cycles.     

Long cycles through prescribed vertices have the Erdős‐Pósa property

We prove that for every graph, any vertex subset S, and given integers : there are k disjoint cycles of length at least ℓ that each contain at least one vertex from S, or a vertex set of size that meets all such cycles. This generalizes previous results of Fiorini and Herinckx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in time, where s denotes the cardinality of S.