Let G and H be graphs. As usual, we say that H is a minor or subcontraction of G if V(G) contains disjoint subsets Wu , u # V(H), such that G[Wu] is connected for each u # V(H) and there is an edgeâ€¦ (More)

A topological complete graph of order p comprises p vertices {v1, . . . , vp} and (p 2 ) pairwise vertex disjoint paths Pi, j , 1 â‰¤ i < j â‰¤ p, such that Pi, j joins vi to v j . It was conjectured byâ€¦ (More)

According to Menger's theorem, if a graph is k-connected then, for any choice of 2k distinct vertices Sl , . . . ,sk, t l , . . . ,tk there are vertex disjoint paths P1,. . . ,Pk joining {Sl , . . .â€¦ (More)

We prove that for every M-dimensional body K, there is a rectangular parallelepiped B of the same volume as K, such that the projection of B onto any coordinate subspace is at most as large as thatâ€¦ (More)

We consider systems where the keys for encrypting messages are derived from the pairwise intersections of sets of private keys issued to the users. We give improved bounds on the storage requirementsâ€¦ (More)

Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G\ the number of vertices. A well-known conjecture of Erdos, related toâ€¦ (More)

Sufficient conditions are given for the existence of an oriented path with given end vertices in a tournament. As a consequence a conjecture of Rosenfeld is established. This states that if n isâ€¦ (More)

We show that a directed graph of order n will contain n-cycles of every orientation, provided each vertex has indegree and outdegree at least (1/2 + n-'")n and n is sufficiently large.