2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 14
author:  Colin J. H. McDiarmid and Tobias Müller 

title:  Colouring random geometric graphs 
keywords:  random geometric graphs, graph colouring 
abstract: 
A random geometric graph
G
is obtained as follows. We take
n
X
at random (i.i.d. according to some probability
distribution
1
, X
2
, …, X
n
∈ℝ
d
ν
on
ℝ
). For
d
i ≠j
we join
X
and
i
X
by an edge if
j
║X
. We study the properties of the chromatic number
i
 X
j
║< r(n)
χ
and clique number
n
ω
of this graph as
n
n
becomes large, where we assume that
r(n) →0
. We allow any choice
ν
that has a bounded density function and
║. ║
may be any norm on
ℝ
. Depending on the choice of
d
r(n)
, qualitatively different types of behaviour can be
observed. We distinguish three main cases, in terms of the
key quantity
n r
(which is a measure of the average degree). If
d
r(n)
is such that
n r
as
d
/
ln
n →0
n →∞
then
χ
almost surely. If
n
/ ω
n
→1
n r
then
d
/
ln
n →∞
χ
almost surely, where
n
/ ω
n
→1 / δ
δ
is the (translational) packing density of
the unit ball
B := { x ∈ℝ
(i.e.
d
: ║x║< 1 }
δ
is the proportion of
d
space that can be filled with disjoint translates of
B
). If
n r
then
d
/
ln
n →t ∈(0,∞)
χ
tends almost surely to a constant that can be bounded
in terms of
n
/ ω
n
δ
and
t
. These results extend earlier work of McDiarmid and
Penrose. The proofs in fact yield separate expressions for
χ
and
n
ω
. We are also able to prove a conjecture by Penrose.
This states that when
n
n r
then the clique number becomes focussed on two
adjacent integers, meaning that there exists a sequence
d
/
ln
n →0
k(n)
such that
P
( ω
n
∈{k(n), k(n)+1}) →1
n →∞
. The analogous result holds for the chromatic number
(and for the maximum degree, as was already shown by
Penrose in the uniform case).

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reference:  Colin J. H. McDiarmid and Tobias Müller (2005), Colouring random geometric graphs, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 14 
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