Trivalent 2-Arc Transitive Graphs of Type G(2)(1) are Near Polygonal

Abstract

A connected graph Σ of girth at least four is called a near n-gonal graph with respect to E, where n ≥ 4 is an integer, if E is a set of n-cycles of Σ such that every path of length two is contained in a unique member of E. It is well known that connected trivalent symmetric graphs can be classified into seven types. In this note we prove that every connected trivalent G-symmetric graph Σ≠K4 of type is a near polygonal graph with respect to two G-orbits on cycles of Σ. Moreover, we give an algorithm for constructing the unique cycle in each of these G-orbits containing a given path of length two. © 2010 Springer Basel AG.