Counting 2-Connected 4-Regular Maps on the Projective Plane

Shude Long, Han Ren

Abstract


In this paper the number of 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, the number of nonroot-vertex-blocks. As special cases, formulae for several types of rooted 4-regular maps such as 2-connected 4-regular projective planar maps, rooted 2-connected (connected) 4-regular projective planar maps without loops are also presented. Several known results on the number of 4-regular maps on the projective plane are also concluded. Finally, by use of Darboux's method, very nice asymptotic formulae for the numbers of those types of maps are given.

Keywords


(Rooted) near-4-regular map; Lagrangian inversion; enumerating function; asymptotic

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