Graph_Theory_Notes8.pdf

# C wikipedia klein bottle non orientable c wikipedia

• Notes
• 9

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c Wikipedia Klein bottle: non-orientable c Wikipedia Every orientable surface is obtained by adding some handles to a sphere. The number of handles added is called the genus of the sur- face. All orientable surfaces of genus γ are “essentially the same”, denoted by S γ . A 2 -cell on a surface is a region such that every closed curve in the interior can be continuously contracted to a point. A 2 -cell embedding of a graph on S γ is an embedding of the graph on S γ such that every region is a 2-cell.

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Euler’s formula for S γ Theorem 14. Let G be a graph with n vertices and m edges. If G can be 2-cell embedded on S γ with f faces, then n - m + f = 2 - 2 γ. Theorem 15. Let G be a graph with n vertices and m edges that can be 2-cell embedded on S γ . Then m 3( n - 2 + 2 γ ) .
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