On the Maximum Number of Vertices of Minimal Embedded Graphs

Consider graphs embedded in a Riemannian manifold, attached at n fixed points. Say such a graph is minimal if it is a critical point of the length functional. In this paper, we give a sharp upper bound for the maximal number of vertices of planar 3-regular minimal graphs. 1 Minimal graphs Definition 1.1 Let M be a Riemannian manifold and let A be a finite subset of M. A minimal graph with attaching points A is a finite embedded graph G in M such that the following conditions are satisfied: I. Each edge of the graph is a geodesic segment II. Every a ∈ A is a vertex of degree 1. III. The sum of unit vectors of edges outcoming from each vertex of degree greater than 1 is equal to zero. Minimal graphs are critical points for the length functional on the space of embedded graphs with fixed attaching points. In this paper we are not interested in the length of graphs. The purpose of our study here is the combinatorial structure of minimal graphs