  Given $0<\alpha\leq\pi$, ${\epsilon}>0$ and $n$, we define random graphs on
the $d$-dimensional sphere by drawing $n$ i.i.d. uniform random points for the
vertices, and edges $u {\sim} v$ whenever the geodesic distance between $u$ and
$v$ is ${\epsilon}$-close to ${\alpha}$. This model generalizes distance graphs
on spheres, and also random Borsuk graphs. Topological tools are known to give
tight bounds for the chromatic number of Borsuk graphs. We now study the
efficiency of one of these topological invariants, namely the connectivity of
L\'ovasz's neighborhood complex, to bound the chromatic number of this model of
random graphs. We show that, in general, this bound performs badly, however, it
still produces some useful bounds in dimensions $d=1$ and 2.
