Given a configuration of indistinguishable pebbles on the vertices of a connected graph G on n vertices, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one pebble on an adjacent vertex. The m-pebbling number of a graph G, πm(G), is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of p...

A graph that contains a spanning tree of diameter at most t clearly admits a tree t-spanner, since a tree t-spanner of a graph G is a sub tree of G such that the distance between pairs of vertices in the tree is at most t times their distance in G. In this paper, graphs that admit a tree t-spanner of diameter at most t + 1 are studied. For t equal to 1 or 2 the problem has been solved. For t = ...

The EAP Re-authentication Protocol (ERP) defines extensions to the Extensible Authentication Protocol (EAP) to support efficient re-authentication between the peer and an EAP Re-authentication (ER) server through a compatible authenticator. This document specifies Diameter support for ERP. It defines a new Diameter ERP application to transport ERP messages between an ER authenticator and the ER...

We prove that for every connected 4-colourable graph G of order n and minimum degree δ ≥ 1, diam(G) ≤ 5n 2δ − 1. This is a first step toward proving a conjecture of Erdős, Pach, Pollack and Tuza [4] from 1989.

Extending some recent theorems of Klee and Larman, we prove rather sharp results about the diameter of a random graph. Among others we show that if d = d(n) > 3 and m = m(n) satisfy (log n)/d 3 log log n -> oo, 2rf_Imd'/'nd+x log n -» oo and dd~2md~l/nd — log n -» -oo then almost every graph with n labelled vertices and m edges has diameter d. About twenty years ago Erdös [7], [8] used random g...

Diameter of the 2 k-node Knn odel graph is d(k + 2)=2e:

For a pair of positive parameters D,χ, a partition P of the vertex set V of an n-vertex graph G = (V,E) into disjoint clusters of diameter at most D each is called a (D,χ) network decomposition, if the supergraph G(P), obtained by contracting each of the clusters of P , can be properly χ-colored. The decompositionP is said to be strong (resp., weak) if each of the clusters has strong (resp., we...

We prove a formula for the Fekete-Leja transfinite diameter of the pullback of a set E ⊂ C by a regular polynomial map F , expressing it in terms of the resultant of the leading part of F and the transfinite diameter of E. We also establish the nonarchimedean analogue of this formula. A key step in the proof is a formula for the transfinite diameter of the filled Julia set of F .

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...