The vertex primitive and vertex bi-primitive s-arc regular graphs

We determine all vertex primitive and vertex bi-primitive 3-arc regular graphs. This result completes the classification of vertex primitive s-arc regular graphs with s ⩾ 2 and the classification of vertex bi-primitive s-arc regular graphs with s ⩾ 3 . In particular, it is shown that the Petersen graph and the Coxeter graph are the only vertex primitive 3-arc regular graphs, and that vertex bi-primitive 3-arc regular graphs consist of the complete bipartite graph K 3 , 3 , the standard double covers of the Petersen graph and the Coxeter graph, two cubic graphs admitting PGL ( 2 , 11 ) and PGL ( 2 , 13 ) , respectively, and a valency four graph admitting P Γ L ( 2 , 27 ) .