Let P be a Poisson process of intensity one in a square
Sn of area n. 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 ≤ 0.074logn
then the probability that Gn,k is
connected tends to 0 as n → ∞ while, if
k ≥ 5.1774logn, then the probability that
Gn,k is connected tends to 1 as
n → ∞. They conjectured that the threshold
for connectivity is k = (1 + o(1))logn. In
this paper we improve these lower and upper bounds to
0.3043logn and 0.5139logn, respectively, disproving
this conjecture. We also establish lower and upper bounds of
0.7209logn and 0.9967logn for the directed version
of this problem. A related question concerns coverage. With
Gn,k as above, we surround each
vertex by the smallest (closed) disc containing its k
nearest neighbours. We prove that if
k ≤ 0.7209logn then the probability that
these discs cover Sn tends to 0 as
n → ∞ while, if
k ≥ 0.9967logn, then the probability that the
discs cover Sn tends to 1 as
n → ∞.
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