Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Thursday, March 16, 2006, 12:15 pm

**Duration**: This information is not available in the database

**Location**: This information is not available in the database

**Speaker**: Tobias Müller (Univ. of Oxford)

A random geometric graph G_{n} is constructed by taking vertices
X_{1}, ..., X_{n} ∈ R^{d} at
random (i.i.d. according to some probability distribution ν) and
including an edge
between X_{i} and X_{j} if |X_{i}-X_{j}| < r where r=r(n) > 0.
We prove a conjecture of Penrose stating that when
n r^{d} = o(ln n) then the probability distribution of the clique number
ω(G_{n}) becomes concentrated on
two consecutive integers in the sense that
P(ω(G_{n}) ∈ {k(n), k(n)+1} ) tends to 1
for some sequence k(n).

We also show that the same holds for a number of other graph parameters
including the chromatic number Χ(G_{n}).
A series of celebrated results establish that a similar phenomenon occurs
in the Erdős-Rényi or G(n,p)-model
of random graphs.

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