Gruber P. Convex and Discrete Geometry

15 Combinatorial Theory of Convex Polytopes 261
4A(T ) + 4π = 2α + 2α + 2β + 2β + 2γ + 2γ, or A(T ) = α + β + γ − π. ⊓⊔
Proof of the polytope formula. We may assume that o ∈ int P. Projecting bd P from
o radially onto S 2 yields a one-fold covering of S 2 by f convex spherical polygons.
Apply the above lemma to each of these polygons. Note that the sum of the interior
angles of such polygons at a common vertex is 2π and that each edge is an edge of
precisely two of these polygons, and add. Then
(1) A(S 2 ) = 4π = 2πv − 2πe + 2π f, or v − e + f = 2.
Since each edge is incident with two vertices and each vertex with at least three
edges, it follows that
3v ≤ 2e, similarly, 3 f ≤ 2e.
By (1),
Adding then yields that
−2v =−2e + 2 f − 4, −2 f =−2e + 2v − 4.
v ≤ 2 f − 4, f ≤ 2v − 4. ⊓⊔
Abstract Graphs, Graphs and Realizations
Before proceeding to the graph version of Euler’s formula, some notation will be
introduced.
A (finite) abstract graph G consists of two sets, the set of vertices V = V(G) =
{1,...,v} and the set of edges E = E(G) which consists of two-element subsets of
V. If{u,w} is an edge, we write also uw for it and call w a neighbour of u and vice
versa. If a vertex is contained in an edge or an edge contains a vertex then the vertex
and the edge are incident. If two edges have a vertex in common, they are incident
at this vertex. A path is a sequence of edges such that each edge is incident with the
preceding edge at one vertex and with the next edge at the other vertex. We may write
a path also as a sequence w1w2 ···wk of vertices. We say that it connects the first
and the last vertex of this sequence. A path is a cycle if the first and the last vertex
coincide. A cycle which consists of k edges is a k-cycle. G is connected if any two
distinct of its vertices are connected by a path. If G has at least k + 1 vertices, it is kconnected
if any two distinct vertices can be connected by k paths which, pairwise,
have only these vertices in common. With some effort it can be shown that this is
equivalent to the following. The graph which is obtained from G by deleting any set
of k − 1 vertices and the edges incident with these, is still connected.
More geometrically, by definition, a (finite) graph consists of a finite point set
in some space, the set of vertices and, for each of a set of pairs of distinct vertices,
of a continuous curve which connects these vertices and contains no vertex in its
relative interior. These curves form the set of edges. The notions and notations which
were defined earlier for abstract graphs are defined for graphs in the obvious way.
Two graphs G, H, one or both of which may be abstract, are isomorphic if there is a