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Trivalent 2-Arc Transitive Graphs of Type G21 are Near Polygonal
Sanming Zhou
Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia
smzhou@ms.unimelb.edu.au
Annals of Combinatorics 14 (3) pp.397-405 September, 2010
AMS Subject Classification: 05C25, 20B25
Abstract:
A connected graph ∑ 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 ∑ 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 ∑≠ K4 of type G21 is a near polygonal graph with respect to two G-orbits on cycles of ∑. Moreover, we give an algorithm for constructing the unique cycle in each of these $G$-orbits containing a given path of length two.
Keywords: symmetric graph; arc-transitive graph; trivalent symmetric graph; near polygonal graph; three-arc graph

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