Medial layer graphs of equivelar 4-polytopes

In any abstract 4-polytope P , the faces of ranks 1 and 2 constitute, in a natural way, the vertices of a medial layer graph G . We prove that when P is finite, self-dual and regular (or chiral) of type { 3 , q , 3 } , then the graph G is finite, trivalent, connected and 3-transitive (or 2-transitive). Given such a graph, a reverse construction yields a poset with some structure (a polystroma); and from a few well-known symmetric graphs we actually construct new 4-polytopes. As a by-product, any such 2- or 3-transitive graph yields at least a regular map (i.e. 3-polytope) of type { 3 , q } .