相关论文: Perfect matchings and Hamiltonian cycles in the pr…

We investigate the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. For all integers $k\geq1$, $r\geq 0$, and $\ell\geq (r+1)r$, and for any…

We study conditions under which a given hypergraph is randomly robust Hamiltonian, which means that a random sparsification of the host graph contains a Hamilton cycle with high probability. Our main contribution provides nearly optimal…

The preferential attachment network with fitness is a dynamic random graph model. New vertices are introduced consecutively and a new vertex is attached to an old vertex with probability proportional to the degree of the old one multiplied…

We study preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014).…

We propose an extension of the preferential attachment scheme by allowing the connecting probability to depend on time t. We estimate the parameters involved in the model by minimizing the expected squared difference between the number of…

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for…

We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known…

We investigate minimum vertex degree conditions for $3$-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections…

We consider models of evolving networks $\{\mathcal{G}_n:n\geq 0\}$ modulated by two parameters: an attachment function $f:\mathbb{N}_0\to\mathbb{R}_+$ and a (possibly random) attachment sequence $\{m_i:i\geq 1\}$. Starting with a single…

For $t \in \mathbb{N}$ and every $i\in[t]$, let $H_i$ be a $d_i$-regular connected graph, with $1<|V(H_i)|\le C$ for some integer $C\ge 2$. Let $G=\square_{i=1}^tH_i$ be the Cartesian product of $H_1, \ldots, H_t$. We show that if $t\ge 5C$…

A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated theorem of Dirac from 1952 asserts that every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to…

We propose an improved algorithm for counting the number of Hamiltonian cycles in a directed graph. The basic idea of the method is sequential acceptance/rejection, which is successfully used in approximating the number of perfect matchings…

A subset $C$ of edges in a $k$-uniform hypergraph $H$ is a \emph{loose Hamilton cycle} if $C$ covers all the vertices of $H$ and there exists a cyclic ordering of these vertices such that the edges in $C$ are segments of that order and such…

For an even integer t \geq 2, the Matchings Connecivity matrix H_t is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph K_t on t vertices; an entry H_t[M_1,M_2] is 1 if M_1\cup M_2 is a…

In this article, we relate the spectrum of the discrete magnetic Laplacian (DML) on a finite simple graph with two structural properties of the graph: the existence of a perfect matching and the existence of a Hamiltonian cycle of the…

Bollob\'as-Riordan random pairing model of a preferential attachment graph $G_m^n$ is studied. Let $\{W_j\}_{j\le mn+1}$ be the process of sums of independent exponentials with mean $1$. We prove that the degrees of the first…

We study the basic preferential attachment process, which generates a sequence of random trees, each obtained from the previous one by introducing a new vertex and joining it to one existing vertex, chosen with probability proportional to…

We study the powers of Hamiltonian cycles in randomly augmented Dirac graphs, that is, $n$-vertex graphs $G$ with minimum degree at least $(1/2+\varepsilon)n$ to which some random edges are added. For any Dirac graph and every integer…

We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham and Holmes. We determine the largest…

In preference modelling, it is essential to determine the number of questions and their arrangements to ask from the decision maker. We focus on incomplete pairwise comparison matrices, and provide the optimal filling in patterns, which…