A complete classification of cubic symmetric graphs of girth 6
Dragan Marušič and Klavdija Kutnar
J. comb. theory, Ser. B Volume 99, Number 1, pp. 162-184, 2009. ISSN 0095-8956

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 View the MathML source-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 View the MathML source, where sset membership, variant{0,1}, tgreater-or-equal, slanted1, and View the MathML source, 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 View the MathML source-cover of one of the following graphs: the cube Q3, the Pappus graph or an View the MathML source-path of order n2/2.

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