A complete classification of cubic symmetric graphs of girth 6

Abstract

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 I k n ( t ) -path, a graph of order either n 2 / 2 or n 2 / 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 r = 3 s p 1 e 1 … p t e t > 3 , where s ∈ { 0 , 1 } , t ⩾ 1 , and p i ≡ 1 ( mod 3 ) , such that X is isomorphic either to a Cayley graph of a dihedral group D 2 r of order 2 r or X is isomorphic to a certain Z r -cover of one of the following graphs: the cube Q 3 , the Pappus graph or an I k n ( t ) -path of order n 2 / 2 .