A characterization of locally finite vertex-transitive graphs (Q1095158)

In 1985 the reviewer asked the following: Is it true if G is a connected, locally-finite graph with the property that all its vertex-deleted subgraphs are isomorphic, then G must be regular? If G is regular then it is easy to prove that its automorphism group must act transitively on its vertex set. It is also not hard to prove that the answer to the above question is yes when G is finite. The author here proves that a locally finite graph without isolated vertices is vertex transitive if and only if all its vertex-deleted subgraphs are isomorphic, thus completely answering the original question. His proof is short and elegant.