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 K3,3, 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 PΓL(2,27).