On the smallest graphs of girth 10 and valency 3

Recently, O'Keefe and Wong have shown that a smallest graph of girth 10 and valency 3 (a (3, 10)-cage) must have 70 vertices. There are three non-isomorphic (3, 10)-cages which have been known for some while. In this paper, it is shown that these are the only three possible (3, 10)-cages. Some of the proofs in this paper are derived with the aid of a computer.