Annals ofCombinatorics I (1997) 55-66 Annals of Combinatorics 0 Springer-Verlag 1997 Enumeration of k-poles 9 Zhicheng Gao and Mizan Rahman Department of Mathematics and Statistics, Carleton University, Ottawa, Canada KI S 5B6 Received November 8, 1996 ,4MS Subject Classification: 05C 10, 05C30 Abstract. A k-pole in this paper is a regular planar map with k vertices. Poles with even degrees were first enumerated by Tutte [9] in 1962 where he obtained a very simple and elegant expres- sion. Using Brown's quadratic method, Bender and Canfield [2] derived two algebraic equations for the generating function of the poles. But the equations seem to be quite complicated for the odd degree case, and so far no progress has been seen in utilizing these equations to derive any result for the number of poles with odd degree. In this paper, we use hypergeometric functions to enumerate poles. We will show that the odd degree case is indeed very different from, and much more complicated than, the even degree case. Keywords: enumeration, pole 1. Introduction In this paper, a map is a connected planar graph embedded in the plane such that no edges cross each other. A map is face (vertex) regular if all its faces (vertices) have the same degree. A map is rooted by specifying a vertex (called the root vertex), an edge incident with the vertex (called the root edge), and a side of the edge. The face on the specified side is called the root face. Two maps are combinatorially equivalent if there is a homomorphism from the plane to itself that carries one map to the other. Two rooted maps are equivalent if the homomorphism also preserves the rooting. The concept of rooting a map was first introduced by Tutte [ 10] in 1960s. He showed that rooting a map destroys the possible symmetries, which makes the enumeration much easier. A map is called face-regular if all its faces have the same degree. A rooted map is near face-regular if all its faces, except possibly its root face, have the same degree. The dual of a face-regular map is a vertex-regular map, which is also called a pole. A k-pole is a pole with k vertices. Let M~k,n be the number of rooted near face- regular maps with n edges and k faces such that each nonroot face has degree d. Then Pk(d) = Mk,kd/2 is the number of k-poles of degree d. The first attack on M~n that we are aware of was made by Tutte [9] in 1962. In that paper, the notion of rooting had * Research supported by NSERC. 55