4-edge-connected 4-regular maps on the projective plane

Abstract

In this paper rooted (near-) 4 -regular maps on the projective plane are investigated with respect to the root-valency, the number of edges, the number of inner faces, the number of nonroot-vertex-loops and the number of separating cycles. In particular, 4 -edge connected 4 -regular maps (which are related to the 3 -flow conjecture by Tutte) are handled. Formulae of several types of rooted 4 -edge-connected 4 -regular maps on the projective plane are presented. Several known results on the number of 4 -regular maps on the projective plane are also derived. Finally, using Darboux’s method, a nice asymptotic formula for the numbers of this type of maps is given which implies that almost every (loopless) 4 -regular map on the projective plane has a separating cycle.