Prove. Let a closed surface of genus p be divided up (i.e. marked up) into a number of regions by means of vertices and arcs i.e. by marking a number of vertices on the surface and joining them by arcs. Then
1) V - E + F = 2 - 2p
V is the number of vertices
E is the number of edges or arcs
F is the number of regions
Proof. A representative form for a surface of genus p is a sphere with p handles. We shall prove the theorem for the case of a sphere S with p handles. Proving it for this case proves it for all closed surfaces of genus p since any other surface of genus p can be continuously deformed into a sphere with p handles and in that deformation the numbers V, E and F won’t change.
We assume our sphere with p handles is overlaid by a network of regions delineated by vertices and edges. Before proceeding with the proof we make two observations important to the proof.
Observation 1. Let us envision a sphere (a sphere with no handles) whose surface has been divided up (marked up) into a network of regions. Each region corresponds to a face. Now let us remove two non-adjacent faces, leaving holes in the surface where the faces were. The sphere has thus been transformed into a surface topologically equivalent to a cylinder. In doing this we have reduced the value of F in the Euler characteristic V - E + F [left member of equation 1) above] by two. This decreases the Euler characteristic by 2. If we were to remove four non-adjacent faces we would reduce the Euler characteristic by 4. Now let us remove 2n non-adjacent faces. We thus create 2n holes in the surface. The value of the Euler characteristic V - E + F for a network-overlaid sphere containing 2n holes is 2 - 2n. This is because a sphere without any missing faces has an Euler characteristic of 2 and if we remove 2n non-adjacent faces from such a sphere (but preserving their boundaries) the value of F in the Euler characteristic decreases by 2n.
Observation 2. The value of the Euler characteristic V - E + F of a cylinder is 0. This is seen by noting that if we have a sphere divided into a network of regions and we remove two non-adjacent regions, the sphere, minus those two regions, has an Euler characteristic of 0 — and such a sphere that is minus two of its regions is topologically equivalent to a cylinder i.e. it can be continuously deformed into a cylinder. [The cylinder has an Euler characteristic of 0 because
the sphere without any missing regions has an Euler characteristic of 2 and taking away two regions decreases the value of F by 2]. Consider the cylinder shown in Fig. 1 divided into regions by the lines AD, BE and CF. It has 3 regions, 6 vertices, and 9 edges and thus its Euler characteristic is V - E + F = 6 - 9 + 3 = 0.
We shall now prove the following assertion: A sphere with 2n holes has the same Euler characteristic as a sphere with n handles (where the sphere with n handles is formed by attaching n cylindrical handles to the holes of a sphere of 2n holes).
Let us start with a network-overlaid sphere containing p handles. Assume that the closed curves A1, A2, B1, B2, ... where the handles join the sphere consist of arcs of the given network. See Fig. 3(c). We lose no generality in this assumption because if it is not so a continuous deformation can make it so. Now let us cut off the handles along curves A1, A2, B1, B2, ... where they join the sphere. We now have a sphere with 2p holes with curves A1, A2, B1, B2, ... delineating the hole boundaries. Assume now, for purposes of illustration, that each hole is bounded by a curve such as that shown in Fig. 2, containing 3 edges and 3 vertices of the network and that the handles are divided into the network of regions shown in Fig. 1. We now put the handles back on the sphere. The question we wish to answer is this: What effect on the Euler characteristic does putting a handle on the sphere have? The answer is that the Euler characteristic is left unchanged, that the Euler characteristic for a sphere with a handle is the same as for the sphere with the holes. When we put the handle on the sphere how have the values of V, E, and F been affected? What we have done is to add 3 regions (the regions ABED, BCFE, and CADF in Fig. 1) and 3 edges (the edges AD, BE and CF) to the network. We have thus increased the values of E and F by equal amounts, 3 units each, and the value of the Euler characteristic V - E + F remains unchanged. We have used this specific case to demonstrate but we could generalize on the logic. The number of edges (or arcs) forming the boundary of the holes could be arbitrary. The number of regions into which the handles are divided could be arbitrary. Thus we see that adding a handle causes no change in the Euler characteristic. Hence we conclude that the Euler characteristic of a network-overlaid sphere with p handles is given by that of a sphere with np holes i.e
V - E + F = 2 - 2p