Trivalent 2-arc transitive graphs of type G 1 2 are near polygonal

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 G 1 2 is a near polygonal graph with respect to two Gorbits 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. Key words: Symmetric graph; arc-transitive graph; trivalent symmetric graph; near polygonal graph; three-arc graph AMS subject classification (2000): 05C25, 20B25