Positional Games are played under several rules on the same hypergraph. We consider some intriguing connection among the outcome of the Maker-Breaker and the Picker-Chooser versions. The later ones were introduced by Beck in [5] and proved to be important in understanding Positional Games in general. Beck had the profound conjecture that playing on the same hypergraph, Picker has better chances...

Let V denote a compact (without boundary) connected Riemannian manifold of dimension n. We denote by K the sectional curvature of V and we set inf K = ird~ K(-r) where 9 runs over all tangent 2-planes in V. One calls V a manifold of non-negative curvature ff inf K>~0. This condition has the following geometric meaning. A n n-dimensional Riemannian manifold has non-negative curvature iff for eac...

Let G = (V,E) be a requirements graph. Let d = (dij)i,j=1 be a length metric. For a tree T denote by dT (i, j) the distance between i and j in T (the length according to d of the unique i − j path in T ). The restricted diameter of T , DT , is the maximum distance in T between pair of vertices with requirement between them. The minimum restricted diameter spanning tree problem is to find a span...

We highlight intriguing analogies between the diameter of a polytope and the largest possible total curvature of the associated central path. We prove continuous analogues of the results of Holt and Klee, and Klee and Walkup: We construct a family of polytopes which attain the conjectured order of the largest curvature, and prove that the special case where the number of inequalities is twice t...

The theory of voltage assignments enables one to construct large graphs (directed as well as undirected) as covering spaces of smaller base graphs. All properties of the large graph, called the lift, are determined by the structure of the base graph and by an assignment of voltages (elements of some group) to its arcs. In this paper we prove several upper bounds on the diameter of the lift in t...

In a peer-to-peer (P2P) network, nodes connect into an existing network and participate in providing and availing of services. There is no dichotomy between a central server and distributed clients. Current P2P networks (e.g., Gnutella) are constructed by participants following their own un-coordinated (and often whimsical) protocols; they consequently suffer from frequent network overload and ...

Concern over fault tolerance in the design of interconnection networks has stimulated interest in finding large graphs with maximum degree ∆ and diameter D such that the subgraphs obtained by deleting any set of s vertices have diameter at most D′, this value being close to D or even equal to it. This is the so-called (∆, D, D′, s)-problem. The purpose of this work has been to study this proble...

Many questions regarding the Tower of Hanoi problem have been posed and answered during the years. Variants of the classical puzzle, such as allowing more than 3 pegs, and imposing limitations on the possible moves among the pegs, raised the analogous questions for those variants. One such question is: given a variant, and a certain number of disks, find a pair of disk arrangements such that th...

The (k − 1)-fault diameter Dk(G) of a k-connected graph G is the maximum diameter of an induced subgraph by deleting at most k − 1 vertices from G. This paper considers the fault diameter of the product graph G1 ∗G2 of two graphs G1 and G2 and proves that Dk1+k2(G1 ∗G2) Dk1(G1)+Dk2(G2)+ 1 if G1 is k1-connected and G2 is k2-connected. This generalizes some known results such as Banič and Žerovni...

Portable devices have more data storage and increasingcommunication capabilities everyday. In addition to classic infras-tructure based communication, these devices can exploit human mo-bility and opportunistic contacts to communicate. We analyze thecharacteristics of such opportunistic forwarding paths. We establishthat opportunistic mobile networks in general are character...