We study various properties of the random planar graph R n , drawn uniformly at random from the class P n of all simple planar graphs on n labelled vertices. In particular, we show that the probability that R n is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability R n has linearly many vertices of a given degree, in each embedding R n has linearly many faces of a given size, and R n has exponentially many automorphisms.
Let ωn denote a random graph with vertex set {1, 2, …, n}, such that each edge is present with a prescribed probability p, independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ωn is, with probability one,
There are $n$ queues, each with a single server. Customers arrive in a
Poisson process at rate $\lambda n$, where $0<\lambda<1$. Upon arrival each
customer selects $d\geq2$ servers uniformly at random, and joins the queue at a
least-loaded server among those chosen. Service times are independent
exponentially distributed random variables with mean 1. We show that the system
is rapidly mixing, and then investigate the maximum length of a queue in the
equilibrium distribution. We prove that with probability tending to 1 as
$n\to\infty$ the maximum queue length takes at most two values, which are
$\ln\ln n/\ln d+O(1)$.Comment: Published at http://dx.doi.org/10.1214/00911790500000710 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Summary.Upper bounds on probabilities of large deviations for sums of bounded independent random variables may be extended to handle functions which depend in a limited way on a number of independent random variables. This 'method of bounded differences' has over the last dozen or so years had a great impact in probabilistic methods in discrete mathematics and in the mathematics of operational research and theoretical computer science. Recently Talagrand introduced an exciting new method for bounding probabilities of large deviations, which often proves superior to the bounded differences approach. In this chapter we introduce and survey these two approaches and some of their applications.
A weighting w of the edges of a graph G induces a colouring of the vertices of G where the colour of vertex v, denoted cv, is e v w(e). We show that the edges of every graph that does not contain a component isomorphic to K2 can be weighted from the set {1, . . . , 30} such that in the resulting vertex-colouring of G, for every edge (u, v) of G, cu = cv.
In cellular telephone networks, sets of radio channels (colors) must be assigned to transmitters (vertices) while avoiding interference. Often, the transmitters are laid out like vertices of a triangular lattice in the plane. We investigated the corresponding weighted coloring problem of assigning sets of colors to vertices of the triangular lattice so that the sets of colors assigned to adjacent vertices are disjoint. We present a hardness result and an efficient algorithm yielding an approximate solution.
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