Tetravalent arc-transitive locally-Klein graphs with long consistent cycles

The topic of this paper is connected tetravalent graphs admitting an arc-transitive group of automorphisms GG, such that the vertex-stabiliser GvGv is isomorphic to the Klein 44-group. Such a graph will be called locally-Klein. A cycle in a graph is said to be consistent if there exists an automorphism of the graph that preserves the cycle set-wise and acts upon it as a one-step rotation. The main result of the paper is a classification of those locally-Klein graphs that contain a consistent cycle of length more than half the order of the graph. As a side result, we define an interesting family of graphs embedded on the torus or on the Klein bottle, such that the automorphism group of the resulting map has two orbits on the edges, two orbits on the vertices and two orbits on the arcs of the graph.