Existence Of Long Cycles Research Articles

Noise sensitivity of critical random graphs

We study noise sensitivity of properties of the largest components \({({\mathcal{C}_j})_{j \geqslant 1}}\) of the random graph \(\mathcal{G}(n,p)\) in its critical window p = (1+λn−1/3)/n. For instance, is the property “\(|{\mathcal{C}_1}|\) exceeds its median size” noise sensitive? Roberts and Şengül (2018) proved that the answer to this is yes if the noise ε is such that ε ≫ n−1/6, and conjectured the correct threshold is ε ≫ n−1/3. That is, the threshold for sensitivity should coincide with the critical window—as shown for the existence of long cycles by the first author and Steif (2015).We prove that for ε ≫ n−1/3 the pair of vectors \({n^{ - 2/3}}{(|{\mathcal{C}_j}|)_{j \geqslant 1}}\) before and after the noise converges in distribution to a pair of i.i.d. random variables, whereas for ε ≪ n−1/3 the ℓ2-distance between the two goes to 0 in probability. This confirms the above conjecture: any Boolean function of the vector of rescaled component sizes is sensitive in the former case and stable in the latter.We also look at the effect of the noise on the metric space \({n^{ - 1/3}}{({\mathcal{C}_j})_{j \geqslant 1}}\). E.g., for ε ≥ n−1/3+o(1), we show that the joint law of the spaces before and after the noise converges to a product measure, implying noise sensitivity of any property seen in the limit, e.g., “the diameter of \({\mathcal{C}_1}\) exceeds its median.”

Super-pancyclic hypergraphs and bipartite graphs

We find Dirac-type sufficient conditions for a hypergraph H with few edges to be hamiltonian. We also show that these conditions guarantee that H is super-pancyclic, i.e., for each A⊆V(H) with |A|≥3, H contains a Berge cycle with vertex set A.We mostly use the language of bipartite graphs, because every bipartite graph is the incidence graph of a multihypergraph. In particular, we extend some results of Jackson on the existence of long cycles in bipartite graphs where the vertices in one part have high minimum degree. Moreover, we prove a conjecture of Jackson from 1981 on long cycles in 2-connected bipartite graphs.

Spectral radius and Hamiltonian properties of graphs, II

In this paper, we first present spectral conditions for the existence of in graphs (2-connected graphs) of order n, which are motivated by a conjecture of Erds. We also prove spectral conditions for the existence of Hamilton cycles in balanced bipartite graphs. This result presents a spectral analog of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs, and extends a previous theorem due to Li and the second author for n sufficiently large. We conclude this paper with two problems on tight spectral conditions for the existence of long cycles of given lengths.

Long Waves In The Pre-Industrial Period: A Time Series Analysis of Soviet Researchers’ Data

Kondratiev long cycle is generally treated as a phenomenon of a modern world economy. However, the existence of major cycles before the Industrial Revolution does not contradict the theoretical views of Kondratiev, the founder of the long-waves theory. We have discovered Kondratiev’s documents, which show him going farther back in history. The key question we are trying to answer is why are major cycles not associated with a pre-industrial economy? We also have at our disposal a number of unknown and little-known historiographical sources, which indicate that Soviet researchers cared about the existence of long cycles in the pre-industrial period. At the same time, Soviet scientists had done a tremendous amount of work to construct the time series on historical data of Russia. Using this data, we concluded that the existence of Kondratiev waves in the Russian Empire was very probable.

Long Cycles in Locally Expanding Graphs, with Applications

We provide sufficient conditions for the existence of long cycles in locally expanding graphs, and present applications of our conditions and techniques to Ramsey theory, random graphs and positional games.

Cycles longs des prix des produits de base : cycles Kondratiev ou cycles Kuznets ?

Les recherches empiriques récentes qui ont été menées sur les super cycles des prix des produits de base nous conduisent, dans cet article, à faire retour aux travaux que N.D. Kondratiev et S. Kuznets consacrèrent à la question des prix dans l’entre-deux-guerres. D’une part, les cycles de longue durée des prix qui avaient été mis en évidence par ces derniers et qui présentaient des périodicités de l’ordre d’une cinquantaine (Kondratiev) et d’une vingtaine d’années (Kuznets) le furent à partir d’une analyse de la dynamique des prix de matières premières. D’autre part, les durées moyennes des super cycles des prix des produits de base qui ont été identifiés au cours de ces dernières années s’étendent sur une séquence temporelle particulièrement large (22 à 54,5 ans) dans laquelle s’insèrent, parmi d’autres, les périodicités Kuznets et Kondratiev. Ce résultat mérite de retenir l’attention car, depuis les années 1980, la plupart des contributions empiriques qui ont corroboré l’existence de cycles longs ont insisté, à une très large majorité, non pas sur la présence de ces deux types de récurrences dans les séries étudiées, mais sur celle de l’une et l’absence de l’autre. A l’encontre de ces études, celles qui ont été menées dernièrement sur les super cycles des prix des produits de base invitent, non pas à opposer Kuznets et Kondratiev, mais, au contraire, à rapprocher ces auteurs sur la question des prix, tous deux étant des précurseurs de la problématique des cycles longs des prix des produits de base.

Bipartite Kneser graphs are Hamiltonian

The Kneser graph K(n,k) has as vertices all k-element subsets of [n]:={1,2,…,n} and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph H(n,k) has as vertices all k-element and (n−k)-element subsets of [n] and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all connected Kneser graphs and bipartite Kneser graphs (apart from few trivial exceptions) have a Hamilton cycle. The main contribution of this work is proving this conjecture for bipartite Kneser graphs. We also establish the existence of long cycles in Kneser graphs (visiting almost all vertices), generalizing and improving upon previous results on this problem.

Long Paths and Cycles Passing Through Specified Vertices Under the Average Degree Condition

Let $G$ be a $k$-connected graph with $k\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree of the vertices other than $x$ and $z$. Further, with this result, we prove that: If $G$ has $n$ vertices and $m$ edges, then it contains a cycle of length at least $2m/(n-1)$ passing through its any $k-1$ specified vertices. Our results generalize a theorem of Fan on the existence of long paths and a classical theorem of Erd\"os and Gallai on the existence of long cycles under the average degree condition.

A scaling approach to the existence of long cycles in Casimir boxes

We analyze the concept of generalized Bose–Einstein condensation (g-BEC), known since 1982 for the perfect Bose gas (PBG) in Casimir (or anisotropic) boxes. Our aim is to establish a relation between this phenomenon and two concepts: the concept of long cycles and the off-diagonal-long-range-order (ODLRO), which are usually considered as an adequate way to describe standard BEC on the ground state for cubic boxes. First, we show that these three criteria are equivalent in this latter case. Then, based on a scaling approach, we revise the formulation of these concepts to prove that the classification of g-BEC into three types I, II, III, implies a hierarchy of long cycles (depending on their size scale) as well as a hierarchy of ODLRO which depends on the coherence length of the condensate.

Finding more perfect matchings in leapfrog fullerenes

We use some recent results on the existence of long cycles in leapfrog fullerenes to establish new exponential lower bounds on the number of perfect matchings in such graphs. The new bounds are expressed in terms of Fibonacci numbers.

An implicit degree condition for long cycles in 2-connected graphs

Let id ( v ) denote the implicit degree of a vertex v . In this work we prove that: If G is a 2-connected graph with max { id ( u ) , id ( v ) } ≥ c / 2 for each pair of nonadjacent vertices u and v that are vertices of an induced claw or an induced modified claw of G , then G contains either a Hamilton cycle or a cycle of length at least c . This extends several previous results on the existence of long cycles in graphs.

A Fan Type Condition For Heavy Cycles in Weighted Graphs

A weighted graph is a graph in which each edge $e$ is assigned a non-negative number $w(e)$, called the weight of $e$. The weight of a cycle is the sum of the weights of its edges. The weighted degree $d^w(v)$ of a vertex $v$ is the sum of the weights of the edges incident with $v$. In this paper, we prove the following result: Suppose $G$ is a 2-connected weighted graph which satisfies the following conditions: 1. $\max\{d^w(x),d^w(y)\mid d(x,y)=2\}\geq c/2$; 2. $w(xz)=w(yz)$ for every vertex $z\in N(x)\cap N(y)$ with $d(x,y)=2$; 3. In every triangle $T$ of $G$, either all edges of $T$ have different weights or all edges of $T$ have the same weight. Then $G$ contains either a Hamilton cycle or a cycle of weight at least $c$. This generalizes a theorem of Fan on the existence of long cycles in unweighted graphs to weighted graphs. We also show we cannot omit Condition 2 or 3 in the above result.

Approximating the Longest Cycle Problem in Sparse Graphs

We consider the problem of finding long paths and cycles in Hamiltonian graphs. The focus of our work is on sparse graphs, e.g., cubic graphs, that satisfy some property known to hold for Hamiltonian graphs, e.g., k-cyclability. We first consider the problem of finding long cycles in 3-connected cubic graphs whose edges have weights $w_i\geq 0$. We find cycles of weight at least ${(\sum w_i^a)}^{\frac{1}{a}}$ for $a=\log_2 3$. Based on this result, we develop an algorithm for finding a cycle of length at least $m^{(\log_3 2)/2}\approx m^{0.315}$ in 3-cyclable graphs with vertices of degree at most 3 and with m edges. As a corollary of this result, for arbitrary graphs with vertices of degree at most 3 that have a cycle of length l (or, more generally, a 3-cyclable minor with degrees at most 3 and with l edges), we find a cycle of length at least $l^{(\log_3 2)/2}$. We consider the graph property of 1-toughness that is common to Hamiltonian graphs and 3-connected cubic graphs, and we try to determine if 1-toughness implies the existence of long cycles. We show that 2-connectivity and 1-toughness, for constant degree graphs, may give cycles that are only of logarithmic length. However, we exhibit a class of 3-connected 1-tough graphs with degrees up to 6, where we can find cycles of length at least ${m}^{\log_3 2}/2$.

A σ_3 type condition for heavy cycles in weighted graphs

A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree dw(v) of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: 1. The weighted degree sum of any three independent vertices is at least m; 2. w(xz)=w(yz) for every vertex z∈N(x)∩ N(y) with d(x,y)=2; 3. In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/3. This generalizes a theorem of Fournier and Fraisse on the existence of long cycles in k-connected unweighted graphs in the case k=2. Our proof of the above result also suggests a new proof to the theorem of Fournier and Fraisse in the case k=2.

Cycles through specified vertices

Recently, various authors have obtained results about the existence of long cycles in graphs with a given minimum degreed. We extend these results to the case where only some of the vertices are known to have degree at leastd, and we want to find a cycle through as many of these vertices as possible. IfG is a graph onn vertices andW is a set ofw vertices of degree at leastd, we prove that there is a cycle through at least\(\left\lceil {\frac{w}{{\left\lceil {{n \mathord{\left/ {\vphantom {n d}} \right. \kern-\nulldelimiterspace} d}} \right\rceil - 1}}} \right\rceil \) vertices ofW. We also find the extremal graphs for this property.

A domain monotonicity theorem for graphs and Hamiltonicity

A necessary condition for the existence of long cycles in a graph involving the Laplacian spectrum of graphs is derived. It is based on two theorems which relate the eigenvalues of a graph and the eigenvalues of its induced subgraphs.