搜索结果：Cycles

An \textit{\(m \times n\) grid graph} is the induced subgraph of the square lattice whose vertex set consists of all integer grid points \(\{(i,j) : 0 \leq i < m,\ 0 \leq j < n\}\). Let $H$ and $K$ be Hamiltonian cycles in an $m \times n$ grid graph $G$. We study the problem of reconfiguring $H$ into $K$, \textcolor{blue}{\textbullet} where the Hamiltonian cycles are viewed as vertices of a reconfiguration graph \textcolor{blue}{\textbullet}, using a sequence of local transformations called \textit{moves}. A \textit{box} of $G$ is a unit square face. A box with vertices $a, b, c, d$ is \textit{switchable} in $H$ if exactly two of its edges belong to $H$, and these edges are parallel. Given such a box with edges $ab$ and $cd$ in $H$, a \textit{switch move} removes $ab$ and $cd$, and adds $bc$ and $ad$. A \textit{double-switch move} consists of performing two consecutive switch moves. If, after a double-switch move, we obtain a Hamiltonian cycle, we say that the double-switch move is \textit{valid}. We prove that any Hamiltonian cycle $H$ can be transformed into any other Hamiltonian cycle $K$ via a sequence of valid double-switch moves, such that every intermediate graph remai

Suppose $G$ is a $k$-uniform hypergraph on $n$ vertices such that every $(k-1)$-subset $S$ of $V(G)$ belongs to at least $δn$ edges, where $δ> 1/2$. Let $Ψ(G)$ denote the number of tight Hamilton cycles in $G$, that is, cyclic orderings of $V(G)$ in which every $k$ consecutive vertices form an edge. We prove that $\logΨ(G)\ge kh(G)-n\log{n\choose k-1}+n\log n-n\log e-o(n)$, where $h(G)$ is the hypergraph entropy of $G$, defined via perfect fractional matchings. This bound is tight, for example, for all (nearly) regular hypergraphs, in particular for the binomial random hypergraph. It also implies a conjecture by Ferber, Hardiman and Mond, stating that $Ψ(G)\ge (δ-o(1))^n n!$.

The generalized Petersen graph $G(n, k)$ is a cubic graph with vertex set $V(G(n, k)) = \{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}$ and edge set $E(G(n, k)) = \{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup \{v_i w_i\}_{0 \leq i < n}$ where the indices are taken modulo $n$. Schwenk found the number of Hamiltonian cycles in $G(n, 2)$, and in this article we present initial conditions and linear recurrence relations for the number of Hamiltonian cycles in $G(n, 3)$ and $G(n, 4)$. This is attained by introducing $G'(n, k)$, which is a modified version of $G(n, k)$, and a subset of its subgraphs which we call admissible, and which are partitioned into different classes in such a manner that we can find relations between the number of admissible subgraphs of each class. The classes and their relations define a directed graph such that each strongly connected component is of a manageable size for $k=3$ and $k=4$, which allows us to find linear recurrence relations for the number of admissible subgraphs in each class in these cases. The number of Hamiltonian cycles in $G(n, k)$ is a sum of the number of admissible subgraphs of $G'(n, k)$ over

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for $r\ge 2$ in every $r$-connected graph any two longest cycles intersect in at least $r$ vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on $n\geq 3$ vertices contains a cycle (and hence path) of length at least $Ω(n^{13/21})$, improving on $Ω(n^{3/5})$ from [DeVos, \emph{arXiv:2302:04255}, 2023]. Second, we show that in every $r$-connected graph with $r\geq 2$, any two longest cycles meet in at least $Ω(r^{5/8})$ vertices, improving on $Ω(r^{3/5})$ from [Chen, Faudree and Gould, \emph{J. Combin. Theory, Ser.~ B}, 1998]. Our proof combines combinatorial arguments, computer-search and linear

We show that for $n \ge 6$ every even permutation on $n$ symbols is the commutator of two $n$-cycles. More precisely, let $S_n$ be the symmetric group and $A_n$ the alternating group. Let $C(n) \subset S_n$ denote the conjugacy class of $n$-cycles and $[\cdot, \cdot]$ be the commutator of two permutations. We prove: The map $C(n) \times C(n) \to A_n, \ (τ, π) \mapsto [τ, π]$ is surjective for all $n \ge 6$.

Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the well-known conjectures of Lovász (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertex-transitive graphs, we present improved bounds for the parameters $\mathrm{lpt}(G)$ and $\mathrm{lct}(G)$, defined as the minimum size of a set of vertices in a graph $G$ hitting all longest paths (cycles, respectively). First, we show that every connected graph $G$ on $n$ vertices satisfies $\mathrm{lpt}(G)\le \sqrt{8n}$, and $\mathrm{lct}(G)\le \sqrt{8n}$ if $G$ is additionally $2$-connected. This improves a sequence of earlier bounds for these problems, with the previous state of the art being $O(n^{2/3})$. Second, we show that every connected graph $G$ satisfies $\mathrm{lpt}(G)\le O(\ell^{5/9})$, where $\ell$ denotes the maximum length of a path in $G$. As an immediate application of this latter bound, we present further progress towards Lovász' and Thomassen's conjectures: We show that every connected vertex-transitive graph of order $n$ contains a cycle (and path) of length $Ω(n^{9/14})$. This improves the previous best bound of the form $Ω(n^{13/21})$. Interesting

We consider the problem of finding a Hamiltonian path or a Hamiltonian cycle with precedence constraints in the form of a partial order on the vertex set. We show that the path problem is $\mathsf{NP}$-complete for graphs of pathwidth 4 while the cycle problem is $\mathsf{NP}$-complete on graphs of pathwidth 5. We complement these results by giving polynomial-time algorithms for graphs of pathwidth 3 and treewidth 2 for Hamiltonian paths as well as pathwidth 4 and treewidth 3 for Hamiltonian cycles. Furthermore, we study the complexity of the path and cycle problems on rectangular grid graphs of bounded height. For these, we show that the path and cycle problems are $\mathsf{NP}$-complete when the height of the grid is greater or equal to 7 and 9, respectively. In the variant where we look for minimum edge-weighted Hamiltonian paths and cycles, the problems are $\mathsf{NP}$-hard for heights 5 and 6, respectively.

For index coding problems with special structure on the side-information graphs called Interlinked Cycle (IC) structures index codes have been proposed in the literature (C. Thapa, L. Ong, and S. Johnson, "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in IEEE Trans. Inf. Theory, vol. 63, no. 6, Jun. 2017, with a correction in "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in arxiv (arxiv:1603.00092v2 [cs.IT] 25 Feb 2018)). Recently (S. Sasi and B.S. Rajan, "On Optimal Index Codes for Interlinked Cycle Structures with Outer Cycles," in arxiv (arXiv:1804.09120v1 [cs.IT]), 24 Apr 2018) for a generalization of IC structures called IC structures with interlocked outer cycles optimal length index codes have been reported and it is shown that the optimal length depends on the maximum number of disjoint outer cycles. In this paper we discuss certain structural properties of IC structures with interlocked outer cycles and provide a simple algorithm to find the maximum number of disjoint outer cycles.

Recurrent boom-and-bust cycles are a salient feature of economic and financial history. Cycles found in the data are stochastic, often highly persistent, and span substantial fractions of the sample size. We refer to such cycles as "long". In this paper, we develop a novel approach to modeling cyclical behavior specifically designed to capture long cycles. We show that existing inferential procedures may produce misleading results in the presence of long cycles, and propose a new econometric procedure for the inference on the cycle length. Our procedure is asymptotically valid regardless of the cycle length. We apply our methodology to a set of macroeconomic and financial variables for the U.S. We find evidence of long stochastic cycles in the standard business cycle variables, as well as in credit and house prices. However, we rule out the presence of stochastic cycles in asset market data. Moreover, according to our result, financial cycles as characterized by credit and house prices tend to be twice as long as business cycles.

We say a graph $H$ decomposes a graph $G$ if there exists a partition of the edges of $G$ into subgraphs isomorphic to $H$. We seek to characterize necessary and sufficient conditions for a cycle of length $k$, denoted $C_k$, to decompose the Cartesian product of two cycles $C_m ~\square~ C_n$. We prove that if $m$ is a multiple of 3, then the Cartesian product of a cycle $C_m$ and any other cycle can be decomposed into 3 cycles of equal length. This extends work of Kotzig, who proved in 1973 that the Cartesian product of two cycles can always be decomposed into two cycles of equal length. We also show that if $k$, $m$, and $n$ are positive, and $k$ divides $4mn$ then $C_{4k}$ decomposes $C_{4m} ~\square~ C_{4n}$.

Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If all adjacent cycles in the sequence share at least one edge, we talk about edge short cycles connectivity. It is shown that the short cycles connectivity is an equivalence relation on the set of vertices, while the edge short cycles connectivity components determine an equivalence relation on the set of edges. Efficient algorithms for determining equivalence classes are presented. Short cycles connectivity can be extended to directed graphs (cyclic and transitive connectivity). For further generalization we can also consider connectivity by small cliques or other families of graphs.

Since its beginnings, every Cycles and Colourings workshop holds one or two open problem sessions; this document contains the problems (together with notes regarding the current state of the art and related bibliography) presented by participants of the 32nd edition of the workshop which took place in Poprad, Slovakia during September 8-13, 2024 (see the workshop webpage https://candc.upjs.sk).

We consider a housing market model with limited externalities where agents care both about their own consumption via demand preferences and about the agent who receives their endowment via supply preferences (we extend the associated lexicographic preference domains introduced in Klaus and Meo, 2023). If preferences are demand lexicographic, then our model extends the classical Shapley-Scarf housing market (Shapley and Scarf, 1974) with strict preferences model. Our main result is a characterization of the corresponding top trading cycles (TTC) rule by individual rationality, pair efficiency, and strategy-proofness (Theorem 1), which extends that of Ekici (2024) from classical Shapley-Scarf housing markets with strict preferences to our model. Two further characterizations are immediately obtained by strengthening pair efficiency to either Pareto efficiency or pairwise stability (Corollaries 1 and 2). Finally, we show that as soon as we extend the preference domain to include demand lexicographic as well as supply lexicographic preferences (e.g., when preferences are separable), no rule satisfying individual rationality, pair efficiency, and strategy-proofness exists (Theorem 2).

The study of variations in solar activity is important for understanding the underlying mechanism of solar activity and for predicting the level of activity in view of the activity impact on space weather and global climate. Here we have used the amplitudes (the peak values of the 13-month smoothed international sunspot number) of Solar Cycles 1-24 to predict the relative amplitudes of the solar cycles during the rising phase of the upcoming Gleissberg cycle. We fitted a cosine function to the amplitudes and times of the solar cycles after subtracting a linear fit of the amplitudes. The best cosine fit shows overall properties (periods, maxima, minima, etc.) of Gleissberg cycles, but with large uncertainties. We obtain a pattern of the rising phase of the upcoming Gleissberg cycle, but there is considerable ambiguity. Using the epochs of violations of the Gnevyshev-Ohl rule (G-O rule) and the `tentative inverse G-O rule' of solar cycles during the period 1610-2015, and also using the epochs where the orbital angular momentum of the Sun is steeply decreased during the period 1600-2099, we infer that Solar Cycle 25 will be weaker than Cycle 24. Cycles 25 and 26 will have almost same

Index code construction for a class of side-information graphs called interlinked cycle (IC) structures without outer cycles is given by Thapa, Ong and Johnson (C. Thapa, L. Ong, and S. Johnson, "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in \textit{IEEE Trans. Inf. Theory, vol. 63, no. 6, Jun. 2017}, "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in arxiv (arxiv:1603.00092v2 [cs.IT] 25 Feb 2018)). In this paper, construction of index codes for interlinked cycle (IC) structures with outer cycles is given along with a decoding algorithm.

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include abelian varieties that have both prime dimension and nontrivial endomorphism ring. The proof is based on a crystalline analogue of Faltings' isogeny theorem due to Bost and the known cases of the Mumford--Tate conjecture.

The authors (Cason, Friedman and Hopkins, Reviews of Economics Studies, 2014) claimed a result that the treatments (using simultaneous matching in discrete time) replicate previous results that exhibit weak or no cycles. After correct two mathematical mistakes in their cycles tripwire algorithm, we research the cycles by scanning the tripwire in the full strategy space of the games and we find significant cycles missed by the authors. So we suggest that, all of the treatments (using simultaneous matching in discrete time) exhibit significant cycles.

We prove some general results about the asymptotics of the distribution of the number of cycles of given length of a random permutation whose distribution is invariant under conjugation. These results were first established to be applied in a forthcoming paper (Cycles of free words in several random permutations with restricted cycles lengths), where we prove results about cycles of random permutations which can be written as free words in several independent random permutations. However, we also apply them here to prove asymptotic results about random permutations with restricted cycle lengths. More specifically, for $A$ a set of positive integers, we consider a random permutation chosen uniformly among the permutations of $\{1,..., n\}$ which have all their cycle lengths in $A$, and then let $n$ tend to infinity. Improving slightly a recent result of Yakymiv (Random A-Permutations: Convergence to a Poisson Process), we prove that under a general hypothesis on $A$, the numbers of cycles with fixed lengths of this random permutation are asymptotically independent and distributed according to Poisson distributions. In the case where $A$ is finite, we prove that the behavior of these

Hajós conjectured in 1968 that every Eulerian \(n\)-vertex graph can be decomposed into at most $\lfloor (n-1)/2\rfloor$ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986), Dean (1986), and Bollobás and Scott (1996) it was analogously conjectured that every \emph{directed} Eulerian graph can be decomposed into $O(n)$ cycles. In this paper, we show that every directed Eulerian graph can be decomposed into $O(n \log Δ)$ disjoint cycles, thus making progress towards the conjecture by Bollobás and Scott. Our approach is based on finding heavy cycles in certain edge-weightings of directed graphs. As a further consequence of our techniques, we prove that for every edge-weighted digraph in which every vertex has out-weight at least $1$, there exists a cycle with weight at least $Ω(\log \log n/{\log n})$, thus resolving a question by Bollobás and Scott.