On Triangulations with High Vertex Degree

We solve three enumerative problems concerning the families of planar maps. More precisely, we establish algebraic equations for the generating function of loopless triangulations in which all vertices have degree at least d, for a certain value d chosen in {3, 4, 5}. The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces. Our proofs first follow Tutte’s classical approach: We decompose maps by deleting the root-edge and translate the decomposition into an equation satisfied by the generating function of the maps under consideration. Then we proceed to solve the equation obtained using a recent technique that extends the so-called quadratic method.