A complete classification of cubic symmetric graphs of girth 6

  • Authors:

  • Klavdija Kutnar;Dragan Marušič

  • Affiliations:

  • University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia;University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia and University of Ljubljana, IMFM, Jadranska 19, 1000 Ljubljana, Slovenia

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2009

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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^sp"1^e^"^1...p"t^e^"^t3, where s@?{0,1}, t=1, and p"i=1(mod3), such that X is isomorphic either to a Cayley graph of a dihedral group D"2"r of order 2r 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.