On narrow hexagonal graphs with a 3-homogeneous suborbit (Q5939548)

A connected graph of girth \(m\geq 3\) is called a polygonal graph if it contains a set of \(m\)-gons with the property that each path of length two belongs to a unique member of the set. A polygonal graph is necessarily regular. If \(m=6\), the graph is called hexagonal. A hexagonal graph \(\Gamma\) is called narrow if the vertices antipodal to a given vertex \(x\) in the hexagons containing \(x\) form an orbit of the stabilizer of \(x\) in \(\text{Aut}(\Gamma)\), and the size of this orbit is \(k-1\), where \(k\) is the valency of \(\Gamma\). The authors classify the narrow hexagonal graphs with vertex-transitive automorphism group, for which the stabilizer of a vertex is 3-homogeneous on the set of neighbors of the vertex (i.e., transitive on the collection of 3-element sets of neighbors).