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LetP be a Poisson process of intensity one in a squareSn of arean. We construct a random geometric graph Gn,k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn,k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that Gn,k is… (More)

- Paul N. Balister, Béla Bollobás, Mark Walters
- Random Struct. Algorithms
- 2005

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 Ac for percolation in this model. The proof is in two parts: first we give a rigorous reduction… (More)

- Paul N. Balister, Béla Bollobás, Amites Sarkar, Santosh Kumar
- MobiCom
- 2007

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)

- Paul N. Balister, Ervin Györi, Jenö Lehel, Richard H. Schelp
- SIAM J. Discrete Math.
- 2007

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 χa(G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χa(G) ≤ 5 for such graphs with maximum… (More)

- Paul N. Balister
- Combinatorics, Probability & Computing
- 2001

- Paul N. Balister
- Combinatorics, Probability & Computing
- 2003

It has been shown [Balister, 2001] that if n is odd and m1, . . . , mt are integers with mi ≥ 3 and ∑t i=1 mi = |E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1, . . . , mt. This result was later generalized [Balister, to appear] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we consider… (More)

- Paul N. Balister, Ervin Györi, Jenö Lehel, Richard H. Schelp
- Combinatorics, Probability & Computing
- 2004

We show that all maximum length paths in a connected circular arc graph have non–empty intersection.

- Paul N. Balister, Béla Bollobás, Alan Stacey
- Random Struct. Algorithms
- 1994

- Paul N. Balister, Béla Bollobás, Stefanie Gerke
- J. Comb. Theory, Ser. B
- 2008

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 Rn chosen uniformly from the set of all graph in A on the vertex set {1, . . . , n}.… (More)

- Paul N. Balister, Jenö Lehel, Richard H. Schelp
- Journal of Graph Theory
- 2006

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 Cn and paths Pn on n vertices are Ramsey unsaturated for all n ≥ 5. 1 Results and… (More)