torus shit
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übung_11/3b.png
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\item
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\item
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Siehe \autoref{fig:1b}.
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Siehe \autoref{fig:1b}.
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\points{1}
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\begin{figure}
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\begin{figure}
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\centering
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\centering
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\includegraphics[page=1, width=0.33\textwidth]{figures.pdf}
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\includegraphics[page=1, width=0.5\textwidth]{figures.pdf}
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\caption{Gegenbeispiel; $K_{3,3}$ hat $9 < 12 = 3 \cdot 6 - 6 $ Kanten und ist nicht planar.}
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\caption{Gegenbeispiel; $K_{3,3}$ hat $9 < 12 = 3 \cdot 6 - 6 $ Kanten und ist nicht planar.}
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\label{fig:1b}
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\label{fig:1b}
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\end{figure}
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\end{figure}
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\newpage
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\section{Planare Graphen}
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\section{Planare Graphen}
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\begin{tasks}
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\item
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O.B.d.A. gilt:
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Angenommen $G$ ist planar. Sei $n = \abs{V}, m = \abs{E}$ und Anzahl der Facetten $f$. Dann gilt
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\begin{align}
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m &\leq 3n - 6 \\
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f &\leq 2n - 4
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\end{align}
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Angenommen $G$ hat $11$ Knoten. Dann ist $m \leq 27 = 3 \cdotp 11 - 6$ und $f \leq 18 = 2 \cdotp 11 - 4$.
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Ein Graph dieser Größe hat maximal $\binom{\abs{V}}{2} = \frac{11 \cdotp 10}{2 \cdotp 1} = 55$ Kanten.
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Angenommen $G$ hat $27$ Kanten. $G'$ hat also $55 - 27 = 28$ Kanten. Das ist ein Widerspruch zum ersten Kriterium. Somit kann $G'$ nicht planar sein.
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\points{3}
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\item
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Siehe \autoref{fig:3b} und \url{https://nevereverever.de/planark7.html}.
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\points{3}
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\begin{figure}
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\centering
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\includegraphics[page=6]{figures.pdf}
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\caption{Planarer $K_7$ auf einem Torus. }
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\label{fig:3b}
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\end{figure}
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\end{tasks}
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übung_11/planark7.blend
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Add table
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Reference in a new issue