In a generalized Maker-Breaker positional game, Maker and Breaker play in turns. Maker makes a moves in each turn and Breaker makes b moves in each turn. We choose Maker to be the first player, although this nearly always makes no difference in the outcome of the game. We call such games (a : b)-games. If a = b, the game is fair, otherwise it is biased. If a = b > 1, the game is accelerated. In...

We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a continues region (Imprecise model) or a finite set of points (Indecisive model). Given a set of inexact points in one of Imprecise or Indecisive models, we wish to provide a lower-bound on the diameter...

A graph is called diameter-k-critical if its diameter is k, and the removal of any edge strictly increases the diameter. In this paper, we prove several results related to a conjecture often attributed to Murty and Simon, regarding the maximum number of edges that any diameter-k-critical graph can have. In particular, we disprove a longstanding conjecture of Caccetta and Häggkvist (that in ever...

Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively.

Given an edge-weighted undirected graph G = (V,E, c, w), where each edge e ∈ E has a cost c(e) and a weight w(e), a set S ⊆ V of terminals and a positive constant D0, we seek a minimum cost Steiner tree where all terminals appear as leaves and its diameter is bounded by D0. Note that the diameter of a tree represents the maximum weight of path connecting two different leaves in the tree. Such p...

The main result of the paper is to prove for every r ≥ 0 and n > n0(r) that every antichain on n vertices and consisting of more than ( n r ) sets contains two members whose symmetric difference is at least 2r + 2. The bound is best possible and all the extremal families are determined. For r ≥ 3 we show that n > 6(r + 1)2 is sufficient.

By Margulis’ result, in our setting diameter and injectivty radius are inversely related. Thus, our theorem can also be viewed as a lower bound on injectivity radius; that is, with the above hypothesis, inj(M) > 1 R(l(P )) . It is known that that infinitely many closed, hyperbolic 3-manifolds of volume less than a given upper bound may be obtained by hyperbolic Dehn surgery on a finite list of ...