4-edge-connected 4-regular maps on the projective plane - Long, Shude (Ars mathematica contemporanea, 2020)
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.
Keywords: (rooted) near-4-regular map; (ukoreninjen) skoraj-4-regularen zemljevid; asimptotika; asymptotic; enumerating function; Lagrangeova inverzija; Lagrangian inversion; ločevalni cikel; preštevalna funkcija; separating cycle