  In this paper, we consider the Voronoi decompositions of an arbitrary
infinite vertex-transitive graph G. In particular, we are interested in the
following question: what is the largest number of Voronoi cells that must be
infinite, given sufficiently (but finitely) many Voronoi sites which are
sufficiently far from each other? We call this number the survival number s(G).
  The survival number of a graph has an alternative characterization in terms
of covering, which we use to show that s(G) is always at least two. The
survival number is not a quasi-isometry invariant, but it remains open whether
finiteness of the s(G) is. We show that all vertex transitive graphs with
polynomial growth have a finite s(G); vertex transitive graphs with infinitely
many ends have an infinite s(G); the lamplighter graph LL(Z), which has
exponential growth, has a finite s(G); and the lamplighter graph LL(Z^2), which
is Liouville, has an infinite s(G).
