diff --git a/presentations/Diskrete-Mathematik/LaTeX/Ende.tex b/presentations/Diskrete-Mathematik/LaTeX/Ende.tex index 0b88ebc..420ecaa 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/Ende.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/Ende.tex @@ -141,8 +141,32 @@ Folge $(G_n)$ aus Graphen abgebildet. Wie sieht $G_4$ aus? \end{gallery} \end{frame} +\begin{frame}{Aufgabe 9, Teil 1 (Lösung)} + \begin{center} + \input{graphs/triangular-4} + \end{center} +\end{frame} + +\begin{frame}{Aufgabe 9, Teil 1 (Lösung)} + \begin{center} + \input{graphs/triangular-5} + \end{center} +\end{frame} + +\begin{frame}{Aufgabe 9, Teil 1 (Lösung)} + \begin{center} + \input{graphs/triangular-6} + \end{center} +\end{frame} + \begin{frame}{Aufgabe 9, Teil 2} -Wieviele Ecken / Kanten hat $G_n = (E_n, K_n)$? +Wie viele Ecken und wie viele Kanten hat $G_i$? + +\begin{gallery} + \galleryimage{graphs/triangular-1} + \galleryimage{graphs/triangular-2} + \galleryimage{graphs/triangular-3} +\end{gallery} \end{frame} \begin{frame}{Aufgabe 9, Teil 2: Antwort} @@ -161,7 +185,32 @@ Kanten: \end{align} \end{frame} +\begin{frame}{Aufgabe 9, Teil 3} +Gebe $G_i$ formal an. +\begin{gallery} + \galleryimage{graphs/triangular-1} + \galleryimage{graphs/triangular-2} + \galleryimage{graphs/triangular-3} +\end{gallery} +\end{frame} + +\begin{frame}{Aufgabe 9, Teil 3 (Lösung)} +Gebe $G_n$ formal an. + +\begin{gallery} + \galleryimage{graphs/triangular-1} + \galleryimage{graphs/triangular-2} + \galleryimage{graphs/triangular-3} +\end{gallery} + +\begin{align*} + E_n &= \Set{e_{x,y} | y \in 1, \dots, n;\; x \in y, \dots, 2 \cdot n - y \text{ mit } x-y \equiv 0 \mod 2}\\ + K_n &= \Set{\Set{e_{x,y}, e_{i,j}} \in E_n^2 | (x+2=i \land y=j) \lor (x+1=i \land y\pm1=j)}\\ + G_n &= (E_n, K_n) +\end{align*} + +\end{frame} \subsection{Bildquelle} \begin{frame}{Bildquelle} diff --git a/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf b/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf index a423a42..d3331e9 100644 Binary files a/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf and b/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf differ diff --git a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-1.tex b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-1.tex index dd5edd4..497ebdb 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-1.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-1.tex @@ -1,19 +1,27 @@ -% A complete graph -% Author: Quintin Jean-Noël -% \documentclass[varwidth=true, border=2pt]{standalone} -\usepackage[nomessages]{fp}% http://ctan.org/pkg/fp +\usepackage{ifthen} \usepackage{tikz} -\usetikzlibrary[topaths] - +\usetikzlibrary{calc} \begin{document} - \newcommand\n{5} - \begin{tikzpicture} - \node[vertex] (N1) at (0,0) {}; - \node[vertex] (N2) at (2,0) {}; - \node[vertex] (N3) at (1,1) {}; +\tikzstyle{vertex}=[draw,red,fill=red,circle, +minimum size=10pt,inner sep=0pt] +\tikzstyle{edge}=[red, very thick] +\begin{tikzpicture} + \newcommand{\n}{1} + \foreach \y in {0, ..., \n}{ + \pgfmathsetmacro{\loopend}{{2*\n-\y}} + \pgfmathsetmacro{\second}{{\y+2}} + \foreach \x in {\y, \second,..., \loopend}{ + \ifthenelse{\n=\y}{\breakforeach}{} + \node (n-\x\y)[vertex] at (\x,\y) {}; - \draw (N1) -- (N3) -- (N2) -- (N1); - \end{tikzpicture} + \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);} + \pgfmathtruncatemacro\X{\x} + \ifthenelse{\X<\loopend}{\draw[edge] (\x,\y) -- (\x+2,\y);}{} + \ifthenelse{\X=\loopend}{}{\draw[edge] (\x,\y) -- (\x+1,\y+1);} + + } + } +\end{tikzpicture} \end{document} diff --git a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-2.tex b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-2.tex index 1f1ab77..a97a4e9 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-2.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-2.tex @@ -1,21 +1,27 @@ -% A complete graph -% Author: Quintin Jean-Noël -% \documentclass[varwidth=true, border=2pt]{standalone} -\usepackage[nomessages]{fp}% http://ctan.org/pkg/fp +\usepackage{ifthen} \usepackage{tikz} -\usetikzlibrary[topaths] - +\usetikzlibrary{calc} \begin{document} - \begin{tikzpicture} - \node[vertex] (N1) at (0,0) {}; - \node[vertex] (N2) at (2,0) {}; - \node[vertex] (N3) at (4,0) {}; - \node[vertex] (N4) at (1,1) {}; - \node[vertex] (N5) at (3,1) {}; - \node[vertex] (N6) at (2,2) {}; +\tikzstyle{vertex}=[draw,red,fill=red,circle, +minimum size=10pt,inner sep=0pt] +\tikzstyle{edge}=[red, very thick] +\begin{tikzpicture} + \newcommand{\n}{2} + \foreach \y in {0, ..., \n}{ + \pgfmathsetmacro{\loopend}{{2*\n-\y}} + \pgfmathsetmacro{\second}{{\y+2}} + \foreach \x in {\y, \second,..., \loopend}{ + \ifthenelse{\n=\y}{\breakforeach}{} + \node (n-\x\y)[vertex] at (\x,\y) {}; - \draw (N1) -- (N4) -- (N2) -- (N5) -- (N4) -- (N6) -- (N5) -- (N3) -- (N2) -- (N1); - \end{tikzpicture} + \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);} + \pgfmathtruncatemacro\X{\x} + \ifthenelse{\X<\loopend}{\draw[edge] (\x,\y) -- (\x+2,\y);}{} + \ifthenelse{\X=\loopend}{}{\draw[edge] (\x,\y) -- (\x+1,\y+1);} + + } + } +\end{tikzpicture} \end{document} diff --git a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-3.tex b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-3.tex index ec783e6..68095ea 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-3.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-3.tex @@ -1,25 +1,27 @@ -% A complete graph -% Author: Quintin Jean-Noël -% \documentclass[varwidth=true, border=2pt]{standalone} -\usepackage[nomessages]{fp}% http://ctan.org/pkg/fp +\usepackage{ifthen} \usepackage{tikz} -\usetikzlibrary[topaths] - +\usetikzlibrary{calc} \begin{document} - \begin{tikzpicture} - \node[vertex] (N1) at (0,0) {}; - \node[vertex] (N2) at (2,0) {}; - \node[vertex] (N3) at (4,0) {}; - \node[vertex] (N4) at (6,0) {}; - \node[vertex] (N5) at (1,1) {}; - \node[vertex] (N6) at (3,1) {}; - \node[vertex] (N7) at (5,1) {}; - \node[vertex] (N8) at (2,2) {}; - \node[vertex] (N9) at (4,2) {}; - \node[vertex] (N10) at (3,3) {}; +\tikzstyle{vertex}=[draw,red,fill=red,circle, +minimum size=10pt,inner sep=0pt] +\tikzstyle{edge}=[red, very thick] +\begin{tikzpicture} + \newcommand{\n}{3} + \foreach \y in {0, ..., \n}{ + \pgfmathsetmacro{\loopend}{{2*\n-\y}} + \pgfmathsetmacro{\second}{{\y+2}} + \foreach \x in {\y, \second,..., \loopend}{ + \ifthenelse{\n=\y}{\breakforeach}{} + \node (n-\x\y)[vertex] at (\x,\y) {}; - \draw (N1) -- (N5) -- (N2) -- (N6) -- (N3) -- (N7) -- (N6) -- (N5) -- (N8) -- (N6) -- (N9) -- (N8) -- (N10) -- (N9) -- (N7) -- (N4) -- (N3) -- (N2) -- (N1); - \end{tikzpicture} + \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);} + \pgfmathtruncatemacro\X{\x} + \ifthenelse{\X<\loopend}{\draw[edge] (\x,\y) -- (\x+2,\y);}{} + \ifthenelse{\X=\loopend}{}{\draw[edge] (\x,\y) -- (\x+1,\y+1);} + + } + } +\end{tikzpicture} \end{document} diff --git a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-4.tex b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-4.tex new file mode 100644 index 0000000..ed93f70 --- /dev/null +++ b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-4.tex @@ -0,0 +1,27 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage{ifthen} +\usepackage{tikz} +\usetikzlibrary{calc} + +\begin{document} +\tikzstyle{vertex}=[draw,red,fill=red,circle, +minimum size=10pt,inner sep=0pt] +\tikzstyle{edge}=[red, very thick] +\begin{tikzpicture} + \newcommand{\n}{4} + \foreach \y in {0, ..., \n}{ + \pgfmathsetmacro{\loopend}{{2*\n-\y}} + \pgfmathsetmacro{\second}{{\y+2}} + \foreach \x in {\y, \second,..., \loopend}{ + \ifthenelse{\n=\y}{\breakforeach}{} + \node (n-\x\y)[vertex] at (\x,\y) {}; + + \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);} + \pgfmathtruncatemacro\X{\x} + \ifthenelse{\X<\loopend}{\draw[edge] (\x,\y) -- (\x+2,\y);}{} + \ifthenelse{\X=\loopend}{}{\draw[edge] (\x,\y) -- (\x+1,\y+1);} + + } + } +\end{tikzpicture} +\end{document} diff --git a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-5.tex b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-5.tex new file mode 100644 index 0000000..2d7bf10 --- /dev/null +++ b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-5.tex @@ -0,0 +1,27 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage{ifthen} +\usepackage{tikz} +\usetikzlibrary{calc} + +\begin{document} +\tikzstyle{vertex}=[draw,red,fill=red,circle, +minimum size=10pt,inner sep=0pt] +\tikzstyle{edge}=[red, very thick] +\begin{tikzpicture} + \newcommand{\n}{5} + \foreach \y in {0, ..., \n}{ + \pgfmathsetmacro{\loopend}{{2*\n-\y}} + \pgfmathsetmacro{\second}{{\y+2}} + \foreach \x in {\y, \second,..., \loopend}{ + \ifthenelse{\n=\y}{\breakforeach}{} + \node (n-\x\y)[vertex] at (\x,\y) {}; + + \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);} + \pgfmathtruncatemacro\X{\x} + \ifthenelse{\X<\loopend}{\draw[edge] (\x,\y) -- (\x+2,\y);}{} + \ifthenelse{\X=\loopend}{}{\draw[edge] (\x,\y) -- (\x+1,\y+1);} + + } + } +\end{tikzpicture} +\end{document} diff --git a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-6.tex b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-6.tex new file mode 100644 index 0000000..0046742 --- /dev/null +++ b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-6.tex @@ -0,0 +1,27 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage{ifthen} +\usepackage{tikz} +\usetikzlibrary{calc} + +\begin{document} +\tikzstyle{vertex}=[draw,red,fill=red,circle, +minimum size=10pt,inner sep=0pt] +\tikzstyle{edge}=[red, very thick] +\begin{tikzpicture} + \newcommand{\n}{6} + \foreach \y in {0, ..., \n}{ + \pgfmathsetmacro{\loopend}{{2*\n-\y}} + \pgfmathsetmacro{\second}{{\y+2}} + \foreach \x in {\y, \second,..., \loopend}{ + \ifthenelse{\n=\y}{\breakforeach}{} + \node (n-\x\y)[vertex] at (\x,\y) {}; + + \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);} + \pgfmathtruncatemacro\X{\x} + \ifthenelse{\X<\loopend}{\draw[edge] (\x,\y) -- (\x+2,\y);}{} + \ifthenelse{\X=\loopend}{}{\draw[edge] (\x,\y) -- (\x+1,\y+1);} + + } + } +\end{tikzpicture} +\end{document} diff --git a/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-7.tex b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-7.tex new file mode 100644 index 0000000..519a4fa --- /dev/null +++ b/presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-7.tex @@ -0,0 +1,27 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage{ifthen} +\usepackage{tikz} +\usetikzlibrary{calc} + +\begin{document} +\tikzstyle{vertex}=[draw,red,fill=red,circle, +minimum size=10pt,inner sep=0pt] +\tikzstyle{edge}=[red, very thick] +\begin{tikzpicture} + \newcommand{\n}{7} + \foreach \y in {0, ..., \n}{ + \pgfmathsetmacro{\loopend}{{2*\n-\y}} + \pgfmathsetmacro{\second}{{\y+2}} + \foreach \x in {\y, \second,..., \loopend}{ + \ifthenelse{\n=\y}{\breakforeach}{} + \node (n-\x\y)[vertex] at (\x,\y) {}; + + \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);} + \pgfmathtruncatemacro\X{\x} + \ifthenelse{\X<\loopend}{\draw[edge] (\x,\y) -- (\x+2,\y);}{} + \ifthenelse{\X=\loopend}{}{\draw[edge] (\x,\y) -- (\x+1,\y+1);} + + } + } +\end{tikzpicture} +\end{document}