Weitere Aufgaben hinzugefügt

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}

presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-1.tex

@@ -1,19 +1,27 @@
-% A complete graph
-% Author: Quintin Jean-Noël
-% <http://moais.imag.fr/membres/jean-noel.quintin/>
 \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}
+\tikzstyle{vertex}=[draw,red,fill=red,circle,
-    \begin{tikzpicture}
+minimum size=10pt,inner sep=0pt]
-        \node[vertex] (N1) at (0,0) {};
+\tikzstyle{edge}=[red, very thick]
-        \node[vertex] (N2) at (2,0) {};
+\begin{tikzpicture}
-        \node[vertex] (N3) at (1,1) {};
+    \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);
+            \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);}
-    \end{tikzpicture}
+            \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}

presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-2.tex

@@ -1,21 +1,27 @@
-% A complete graph
-% Author: Quintin Jean-Noël
-% <http://moais.imag.fr/membres/jean-noel.quintin/>
 \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}
+\tikzstyle{vertex}=[draw,red,fill=red,circle,
-        \node[vertex] (N1) at (0,0) {};
+minimum size=10pt,inner sep=0pt]
-        \node[vertex] (N2) at (2,0) {};
+\tikzstyle{edge}=[red, very thick]
-        \node[vertex] (N3) at (4,0) {};
+\begin{tikzpicture}
-        \node[vertex] (N4) at (1,1) {};
+    \newcommand{\n}{2}
-        \node[vertex] (N5) at (3,1) {};
+    \foreach \y in {0, ..., \n}{
-        \node[vertex] (N6) at (2,2) {};
+        \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);
+            \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);}
-    \end{tikzpicture}
+            \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}

presentations/Diskrete-Mathematik/LaTeX/graphs/triangular-3.tex

@@ -1,25 +1,27 @@
-% A complete graph
-% Author: Quintin Jean-Noël
-% <http://moais.imag.fr/membres/jean-noel.quintin/>
 \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}
+\tikzstyle{vertex}=[draw,red,fill=red,circle,
-        \node[vertex] (N1) at (0,0) {};
+minimum size=10pt,inner sep=0pt]
-        \node[vertex] (N2) at (2,0) {};
+\tikzstyle{edge}=[red, very thick]
-        \node[vertex] (N3) at (4,0) {};
+\begin{tikzpicture}
-        \node[vertex] (N4) at (6,0) {};
+    \newcommand{\n}{3}
-        \node[vertex] (N5) at (1,1) {};
+    \foreach \y in {0, ..., \n}{
-        \node[vertex] (N6) at (3,1) {};
+        \pgfmathsetmacro{\loopend}{{2*\n-\y}}
-        \node[vertex] (N7) at (5,1) {};
+        \pgfmathsetmacro{\second}{{\y+2}}
-        \node[vertex] (N8) at (2,2) {};
+        \foreach \x in {\y, \second,..., \loopend}{
-        \node[vertex] (N9) at (4,2) {};
+            \ifthenelse{\n=\y}{\breakforeach}{}
-        \node[vertex] (N10) at (3,3) {};
+            \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);
+            \ifthenelse{\y=0}{}{\draw[edge] (\x,\y) -- (\x+1,\y-1);}
-    \end{tikzpicture}
+            \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}

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}

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}

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}

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}