Abstract
A connected graph of girth i>m ≥ 3 is called a polygonal graph if it contains a set of i>m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showed that the study of such graphs divides naturally into a number of substantial subcases. Here we analyze one of these cases and characterize the i>k-valent polygonal graphs of girth 6 which have automorphism groups transitive on vertices, which preserve the set of special hexagons, and which have a suborbit of size i>k − 1 at distance three from a given vertex.
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