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In 1961 Gilbert defined a model of continuum percolation in which points are placed in the plane according to a Poisson process of density one, and two are joined if one lies within a disc of area A about the other. We prove some good bounds on the critical area A c for percolation in this model. The proof is in two parts: first we give a rigorous reduction… (More)

A non-empty class A of labelled graphs is weakly addable if for each graph G ∈ A and any two distinct components of G, any graph that can be obtain by adding an edge between the two components is also in A. For a weakly addable graph class A, we consider a random element R n chosen uniformly from the set of all graph in A on the vertex set {1,. .. , n}.… (More)

We model the dynamical behavior of the neuropil, the densely interconnected neural tissue in the cortex, using neuropercolation approach. Neuropercolation generalizes phase transitions modeled by percolation theory of random graphs, motivated by properties of neurons and neural populations. The generalization includes (1) a noisy component in the… (More)

It has been shown [Balister, 2001] that if n is odd and m 1 ,. .. , mt are integers with m i ≥ 3 and t i=1 m i = |E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m 1 ,. .. , mt. This result was later generalized [Balister, to appear] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we… (More)

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some conjectures and results concerning both Ramsey saturated and unsaturated graphs. In particular, we show that cycles C n and paths P n on n vertices are Ramsey unsaturated for all n ≥ 5. the minimum… (More)

Deriving the critical density (which is equivalent to deriving the critical radius or power) to achieve coverage and/or connectivity for random deployments is a fundamental problem in the area of wireless networks. The probabilistic conditions normally derived, however, have limited appeal among practitioners because they areoften asymptotic, i.e., they… (More)

—Tracking of movements such as that of people, animals, vehicles, or of phenomena such as fire, can be achieved by deploying a wireless sensor network. So far only prototype systems have been deployed and hence the issue of scale has not become critical. Real-life deployments, however, will be at large scale and achieving this scale will become… (More)

We consider lower bounds on the the vertex-distinguishing edge chromatic number of graphs and prove that these are compatible with a conjecture of Burris and Schelp [8]. We also find upper bounds on this number for certain regular graphs G of low degree and hence verify the conjecture for a reasonably large class of such graphs.

An adjacent vertex distinguishing edge-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors χ a (G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χ a (G) ≤ 5 for such graphs with… (More)