Does anyone know if there is a result that relates a quantity such as an average degree to the fact the a (simple and connected graph) has no cut vertices?
e.g. if a graph has a Hamiltoninan cycle then it has not cut vertex. By Ore's theorem, if deg(v) >= n/2 for each vertex v (n is the number of vertices of a graph), then the graph is Hamiltoninan. Hence deg(v) >= n/2 implies no cut vertices (non-separable).
I am looking for a result similar to Ore's theorem, but w.r.t. non-separability e.g. something like if the deg(v) >= f(n) for each v (or some statement about the average degree), then the number of cut vertices is zero.