We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network G admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-...

A vertex set S ⊆ V (G) is k-weak-edge-connected if for every C ⊂ S and x ∈ S−C there are min{k, |C|} edge-disjoint (x,C)-paths in G. For a graph G and a k-weakedge-connected vertex set S ⊂ V (G) with k ≥ 3 and 4 ≤ |S| ≤ 2k, we show that G has an eulerian subgraph containing all vertices in S.

A digraph is locally semicomplete if for every vertex x, the set of in-neighbors as well as the set of out-neighbors of x induce semicomplete digraphs. Let D be a k-connected locally semicomplete digraph with k ≥ 3 and g denote the length of a longest induced cycle of D. It is shown that if D has at least 7(k− 1)g vertices, then D has a factor composed of k cycles; furthermore, if D is semicomp...

We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we improve and generalize the coupling method introduced for random intersection graphs so that it may be used for a wider range of parameters. Using the new approach we are able to tighten the best known results concerning random intersection graphs and establish threshold functio...

The odd graph Ok has the subsets with k elements of a set {1, . . . , 2k + 1} as its vertices set, and there exists an edge between two vertices if the corresponding pair of k-subsets is disjoint. A conjecture claims that Ok is hamiltonian for k > 2 and another long-standing conjecture implies that all odd graphs have a hamiltonian path. We proved that the prism over Ok is hamiltonian and that ...

We describe a general solution method for the problem of finding the shortest path between two vertices of a graph in which each edge has some transit time, costs can vary with time, and stopping and parking (with corresponding costs) are allowed at the vertices.

It is known that every hypercube Qn is a bipartite graph. Assume that n 2 and F is a subset of edges with |F | n− 2. We prove that there exists a hamiltonian path in Qn − F between any two vertices of different partite sets. Moreover, there exists a path of length 2n − 2 between any two vertices of the same partite set. Assume that n 3 and F is a subset of edges with |F | n− 3. We prove that th...

For 1 ≤ ` < k, an `-overlapping cycle is a k-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of k consecutive vertices and every two consecutive edges share exactly ` vertices. A k-uniform hypergraph H is `-Hamiltonian saturated, 1 ≤ ` ≤ k − 1, if H does not contain an `-overlapping Hamiltonian cycle C (k) n (`) but every hypergraph obtained from H by adding on...

Let C m T denote the Kronecker product of a cycle C m and a tree T. If m is odd, then C m T is connected, otherwise this graph consists of two isomorphic components. This paper presents a scheme which constructs a long cycle in each component of C m T, where T satisses certain degree constraints. The cycle thus traced is shown to be a dominating set, and in some cases, a vertex cover of that co...

We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u) + d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u) + d(v) ≥ n + z, where z = 0 when n is odd and z = 1 otherwise, then G contains a cycle of length m ...