Enumeration of 4-regular one-face maps

Enumeration of planar two-face maps

We enumerate unrooted planar maps (up to orientation preserving homeomorphism) having two faces, according to the number of vertices and to their vertex and face degree distributions, both in the (vertex) labelled and unlabelled cases. We rst consider plane maps, i.e., maps which are embedded in the plane, and then deduce the case of planar (or sphere) maps, embedded on the sphere. A crucial st...

Enumeration of planar 2 - face maps 1

Up to now, most of the work on maps has dealt with rooted maps, that is, maps with a distinguished and directed edge. We get rid of this restriction in the case of planar maps having two faces. We enumerate these maps according to their vertex and face degree distributions. The following classes of non rooted 2-face maps are treated: (vertex) labelled or unlabelled, embedded in the plane or on ...

Enumeration of r-regular Maps on the Torus. Part II: Enumeration of Unsensed Maps

Enumeration of labeled 4-regular planar graphs

In this extended abstract, we present the first combinatorial scheme for counting labeled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function counting labeled 4regular planar graphs can be computed effectively as the solution of a system of equations. From here we can extract the coefficients by means of algebraic ...

Enumeration of unrooted odd-valent regular planar maps

We derive closed formulae for the numbers of rooted maps with a fixed number of vertices of the same odd degree except for the root vertex and one other exceptional vertex of degree 1. The same applies to the generating functions for these numbers. Similar results, but without the vertex of degree 1, were obtained by the first author and Rahman. We also show, by manipulating a recursion of Bout...