On Alternatively Connected Edge-Transitive Graphs of Square-free Order.

An edge-transitive graph $$\varGamma $$ is called alternatively connected if a subgroup G of the automorphism group of $$\varGamma $$ has two orbits on the arc set of $$\varGamma $$ , and there exists an alternative walk (with respect to a given G-orbit on arcs) between every pair of vertices of $$\varGamma $$ . Employing the standard double covers of digraphs, we give some basic properties of alternatively connected edge-transitive graphs. The main result of this paper is a reduction result on alternatively connected edge-transitive graphs of square-free order. As an application of this result, we give a characterization for alternatively connected edge-transitive graphs of square-free order and valency 6. It is proved that such a graph is either a circulant or constructed from $$\mathrm{PSL}(2,p)$$ . [ABSTRACT FROM AUTHOR]