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Kantenzug-Defnition verbessert; Definition einer Schleife hinzugefügt; RectangleFreeColoring; Hierholzer-Algorithmus; Nicht-Eindeutigkeit von Eulerkreisen

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Martin Thoma 2013-06-13 20:59:55 +02:00
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presentations/Diskrete-Mathematik/LaTeX/Ende.tex
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@ -212,6 +212,66 @@ Gebe $G_n$ formal an.
\end{frame}
\begin{frame}{{\sc RectangleFreeColoring}}
\begin{block}{{\sc RectangleFreeColoring}}
Gegeben ist $n, m \in \mathbb{N}_{\geq 1}$ und ein
ungerichteter Graph $G = (E, K)$ mit
\[E = \Set{e_{x,y} | 1 \leq x \leq n} \land 1 \leq y \leq m\]
und
\[K = \Set{k=\Set{e_{x,y}, e_{x',y'}} \in E \times E : |x-x'| + |y-y'| = 1} \]
Färbe die Ecken von $G$ min einer minimalen Anzahl von Farben so, dass gilt:
\[\forall e_{x,y}, e_{x',y'} \in E: \neg(c(e_{x,y}) = c(e_{x',y'}) = c(e_{x',y}) = c(e_{x,y'}))\]
\end{block}
\end{frame}
\begin{frame}{{\sc RectangleFreeColoring}}
$4 \times 4$ - Instanz:\\
\vspace{1cm}
\begin{tikzpicture}
\newcommand{\n}{4}
\newcommand{\m}{4}
\foreach \x in {1, ..., \n}{
\foreach \y in {1, ..., \m}{
\node[vertex] (n-\x-\y) at (\x,\y) {};
}
}
\foreach \x in {1, ..., \n}{
\foreach \y in {1, ..., \m}{
\ifthenelse{\x<\n}{\draw (\x,\y) -- (\x+1,\y);}{}
}
}
\foreach \y in {1, ..., \m}{
\foreach \x in {1, ..., \n}{
\ifthenelse{\y<\m}{\draw (\x,\y) -- (\x,\y+1);}{}
}
}
\node[vertex,blue] (n-1-1) at (1,1) {};
\node[vertex,blue] (n-2-1) at (2,1) {};
\node[vertex,blue] (n-3-1) at (3,1) {};
\node[vertex,red] (n-4-1) at (4,1) {};
\node[vertex,blue] (n-1-1) at (1,2) {};
\node[vertex,red] (n-2-1) at (2,2) {};
\node[vertex,red] (n-3-1) at (3,2) {};
\node[vertex,blue] (n-4-1) at (4,2) {};
\node[vertex,red] (n-1-1) at (1,3) {};
\node[vertex,blue] (n-2-1) at (2,3) {};
\node[vertex,red] (n-3-1) at (3,3) {};
\node[vertex,blue] (n-4-1) at (4,3) {};
\node[vertex,red] (n-1-1) at (1,4) {};
\node[vertex,red] (n-2-1) at (2,4) {};
\node[vertex,blue] (n-3-1) at (3,4) {};
\node[vertex,blue] (n-4-1) at (4,4) {};
\end{tikzpicture}
\end{frame}
\subsection{Bildquellen}
\begin{frame}{Bildquellen}
\begin{itemize}