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4-edge-connected 4-regular maps on the projective plane. (English) Zbl 1444.05047
Summary: 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.
MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C30 Enumeration in graph theory
05C45 Eulerian and Hamiltonian graphs
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