Symmetric Cubic Graphs of Girth at Most 7

By a symmetric graph we mean a graph X which automorphism group acts transitively on the arcs of X. A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. Tutte [31, 32] showed that every finite symmetric cubic graph is s-regular for some s ≤ 5. It is well-known that there are precisely five symmetric cubic graphs of girth less than 6. All these graphs can be represented as one-skeletons of regular polyhedra in the plane, projective plane or in torus. With the exception of K3,3, we can find an associated regular polyhedron such that the girth of the graph coincide with the face-size. In this paper we show that with three more exceptions the symmetric cubic graphs of girth g ≤ 7 are one-skeletons of trivalent regular maps with face-size g. All the symmetric cubic graphs of girth 6 except the generalised Petersen graphs GP (8, 3) and GP (10, 3) are one-skeletons of toroidal regular maps of type {6, 3}. We give a simple numerical criterium to determine the degree s of s-regularity of these graphs and derive the presentations of the automorphism groups. As concerns girth 7, the only exceptional graph is the well-known Coxeter graph on 28 vertices. We prove that all the other symmetric cubic graphs of girth 7 are underlying graphs of regular maps of type {7, 3} which are known as Hurwitz maps. Some more results on symmetric cubic graphs with exactly two girth cycles passing through an edge are proved