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 \({\Sigma \neq K_4}\) of type \({G^1_2}\) 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.
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Supported by a Discovery Project Grant (DP0558677) of the Australian Research Council and a Melbourne Early Career Researcher Grant of The University of Melbourne.
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Zhou, S. Trivalent 2-Arc Transitive Graphs of Type \({G^1_2}\) are Near Polygonal. Ann. Comb. 14, 397–405 (2010). https://doi.org/10.1007/s00026-010-0066-1
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DOI: https://doi.org/10.1007/s00026-010-0066-1