A complete classification of cubic symmetric graphs of girth 6

A complete classification of cubic symmetric graphs of girth 6 is given. It is shown that with the exception of the Heawood graph, the Moebius–Kantor graph, the Pappus graph, and the Desargues graph, a cubic symmetric graph X of girth 6 is a normal Cayley graph of a generalized dihedral group; in particular,(i)X is 2-regular if and only if it is isomorphic to a so-called -path, a graph of order either n2/2 or n2/6, which is characterized by the fact that its quotient relative to a certain semiregular automorphism is a path.(ii)X is 1-regular if and only if there exists an integer r with prime decomposition , where s∈{0,1}, t⩾1, and , such that X is isomorphic either to a Cayley graph of a dihedral group D2r of order 2r or X is isomorphic to a certain Zr-cover of one of the following graphs: the cube Q3, the Pappus graph or an -path of order n2/2.