The Euler characteristic of any sphere is at most 2.
Proof: We can define Euler characteristic of a network N to be X(N) =V-E, because a network as such has no faces. Only when it is drawn on a surface it is possible to define faces.
Claim: X(N) < or = 1
If N has a circuit, we break it by throwing away an edge. This decreases E and so increases X(N). We repeat this until no loops are left. A network without any loop is called a tree. On the tree that we have got, we apply collapses by removing a vertex from the end of a branch and also the edge attached. The result leaves the Euler characteristic unchanged. After applying the necessary number of collapses, the network reduces to a point for which the Euler characteristic is 1. This proves that X(N)<or=1.
Note also that the Euler characteristic of any tree is exactly 1. Now start with a given triangulation of S. we define a new triangulation called dual triangulation as follows. Put a vertex in the middle of each triangle, and draw an edge in two vertices in adjacent triangles. The vertices and edges of the dual triangulation form a network. Inside the network we can find some trees (for example, a single vertex). Among these, look at the maximal tree, call it the maximal dual tree. It is easy to see by maximality that, the maximal dual tree must contain all the vertices of the dual triangulation. Let M be the maximal dual tree and let C be that part of the original network comprising the original vertices, together with the edges that do not meet M. Then there are bijections between
This means that X(S)= X(M) + X(C). Since M is a tree X(M)=1 and X(C) <or=1. Thus, X(S)<or=2 .