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## Appendix APreliminary

### A.1 Fundamental theorems

In this chapter, we present some fundamental theorems that are used in the book. Note that the proofs of some theorems are not presented here if they can be found in some popular graph theory textbooks (for instance, [15], [18], [19], [34], [41], [92], and [242], etc.).

#### * Embedding

Theorem A.1.1(Euler formula) Let G = (V, E) be a connected graph with a 2-cell embedding on a surface ∑. Then

where F(G) is the set of faces of G on ∑, ξ(∑) is the Euler characteristic of the surfacesuch that

(1) ξ(∑) ≤ 2 for any surface ∑,

(2) ξ(∑) = 2 ifis a sphere,

(3) ξ(∑) = 1 ifis a projective plane,

(4) ξ(∑) ...

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