  We revisit the classical hard-core model, also known as independent set and
dual to vertex cover problem, where one puts particles with a first-neighbor
hard-core repulsion on the vertices of a random graph. Although the case of
random graphs with small and very large average degrees respectively are quite
well understood, they yield qualitatively different results and our aim here is
to reconciliate these two cases. We revisit results that can be obtained using
the (heuristic) cavity method and show that it provides a closed-form
conjecture for the exact density of the densest packing on random regular
graphs with degree K>=20, and that for K>16 the nature of the phase transition
is the same as for large K. This also shows that the hard-code model is the
simplest mean-field lattice model for structural glasses and jamming.
