diff --git a/school/di-ma/uebung/04/04_1.tex b/school/di-ma/uebung/04/04_1.tex
new file mode 100644
index 0000000..6a1c396
--- /dev/null
+++ b/school/di-ma/uebung/04/04_1.tex
@@ -0,0 +1,70 @@
+\documentclass[12pt,a4paper,german]{article}
+\usepackage{url}
+%\usepackage{graphics}
+\usepackage{times}
+\usepackage[T1]{fontenc}
+\usepackage{ngerman}
+\usepackage{float}
+\usepackage{diagbox}
+\usepackage[utf8]{inputenc}
+\usepackage{geometry}
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{csquotes}
+\usepackage{graphicx}
+\usepackage{epsfig}
+\usepackage{paralist}
+\usepackage{tikz}
+\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
+
+%%%%%%%%%% Fill out the the definitions %%%%%%%%%
+\def \name {Valentin Brandl}					%
+\def \matrikel {108018274494}				%
+\def \pname {Marvin Herrmann}				%
+\def \pmatrikel {108018265436}				%
+\def \gruppe {2 (Mi 10-12 Andre)}
+\def \qname {Pascal Brackmann}
+\def \qmatrikel {108017113834}				%
+\def \uebung {4}								%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ % DO NOT MODIFY THIS HEADER
+\newcommand{\hwsol}{
+\vspace*{-2cm}
+\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
+\noindent \pmatrikel \quad \pname \\
+\noindent \qmatrikel \quad \qname \\
+\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
+}
+
+\begin{document}
+%Import header
+\hwsol
+
+\section*{Aufgabe 4.1}
+\begin{enumerate}[i)]
+
+    \item Je zwei Koten in $G$ sind durch genau einen Pfad miteinander verbunden.
+
+        $\Rightarrow G$ ist zusammenhängend $\Rightarrow G$ ist ein Baum (Lemma b) aus der Vorlesung)
+
+        $\Rightarrow$ i) $\equiv$ iii)
+
+    \item $G$ ist zusammenhängend und es gilt $|V| = |E| + 1$
+
+
+    \item $G$ besitzt keinen einfachen Kreis und es gilt $|V| = |E| + 1$
+
+        $|V| = |E| + 1 \Rightarrow |V| - |E| = 1 \Rightarrow G$ hat eine Zusammenhangskomponente $\Rightarrow G$ ist
+        zusammenhängend.
+
+        Kreisfrei und zusammenhängend $\Rightarrow G$ ist ein Baum
+
+        $\Rightarrow$ ii) $\equiv$ iii)
+
+\end{enumerate}
+
+Da i) $\equiv$ iii) und ii) $\equiv$ iii) $\Rightarrow$ i) $\equiv$ ii) $\equiv$ iii)
+
+\end{document}
+
diff --git a/school/di-ma/uebung/04/04_2.tex b/school/di-ma/uebung/04/04_2.tex
new file mode 100644
index 0000000..de2f991
--- /dev/null
+++ b/school/di-ma/uebung/04/04_2.tex
@@ -0,0 +1,129 @@
+\documentclass[12pt,a4paper,german]{article}
+\usepackage{url}
+%\usepackage{graphics}
+\usepackage{times}
+\usepackage[T1]{fontenc}
+\usepackage{ngerman}
+\usepackage{float}
+\usepackage{diagbox}
+\usepackage[utf8]{inputenc}
+\usepackage{geometry}
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{csquotes}
+\usepackage{graphicx}
+\usepackage{epsfig}
+\usepackage{paralist}
+\usepackage{tikz}
+\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
+
+%%%%%%%%%% Fill out the the definitions %%%%%%%%%
+\def \name {Valentin Brandl}					%
+\def \matrikel {108018274494}				%
+\def \pname {Marvin Herrmann}				%
+\def \pmatrikel {108018265436}				%
+\def \gruppe {2 (Mi 10-12 Andre)}
+\def \qname {Pascal Brackmann}
+\def \qmatrikel {108017113834}				%
+\def \uebung {4}								%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ % DO NOT MODIFY THIS HEADER
+\newcommand{\hwsol}{
+\vspace*{-2cm}
+\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
+\noindent \pmatrikel \quad \pname \\
+\noindent \qmatrikel \quad \qname \\
+\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
+}
+
+\begin{document}
+%Import header
+\hwsol
+
+\section*{Aufgabe 4.2}
+\begin{enumerate}[1.]
+
+    \item Graph $G=(V,E)$ mit\\
+        Knotenmenge: $V=\{a,b,c,d,e,f\}$\\
+        Kantenmenge: $E=
+        \{
+            \{a,b\},
+            \{a,c\},
+            \{a,d\},
+            \{a,f\},
+            \{b,f\},
+            \{b,e\},
+            \{c,d\},
+            \{d,e\}
+        \}
+        $
+        \\
+        Adjazenz-Matrix:\\
+        \begin{tabular}{c|ccccccc}
+        & a & b & c & d & e &  f \\
+        \hline
+            a & 0 & 1 & 1 & 1 & 0 & 1 \\
+            b & 1 & 0 & 0 & 0 & 1 & 1 \\
+            c & 1 & 0 & 0 & 1 & 0 & 0 \\
+            d & 1 & 0 & 1 & 0 & 1 & 0 \\
+            e & 0 & 1 & 0 & 1 & 0 & 0 \\
+            f & 1 & 1 & 0 & 0 & 0 & 0 \\
+        \end{tabular}
+        \\
+        \\
+        Inzidenz-Matrix:\\
+        \begin{tabular}{c|ccccccccc}
+            a & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
+            b & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
+            c & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\
+            d & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\
+            e & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\
+            f & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
+        \end{tabular}
+
+    \item \begin{tikzpicture}
+        \def \n {5}
+        \def \radius {3cm}
+        \def \margin {8} % margin in angles, depends on the radius
+        \node[draw, circle] (a) at (3,2) {a};
+        \node[draw, circle] (b)  at (1,1) {b};
+        \node[draw, circle] (c) at (2,1) {c};
+        \node[draw, circle] (d)  at (3,1) {d};
+        \node[draw, circle] (f)  at (4,1) {f};
+        \node[draw, circle] (e)  at (0,0) {e};
+
+        \draw (a)--(b);
+        \draw (a)--(c);
+        \draw (a)--(d);
+        \draw (a)--(f);
+        \draw (b)--(e);
+        \end{tikzpicture}
+        \\
+        $G \setminus \{\{b,f\},\{c,d\},\{d,e\}\}$
+
+        \begin{tikzpicture}
+        \def \n {5}
+        \def \radius {3cm}
+        \def \margin {8} % margin in angles, depends on the radius
+        \node[draw, circle] (b) at (2,2) {b};
+        \node[draw, circle] (a)  at (1,1) {a};
+        \node[draw, circle] (f) at (2,1) {f};
+        \node[draw, circle] (e)  at (3,1) {e};
+        \node[draw, circle] (c)  at (0,0) {c};
+        \node[draw, circle] (d)  at (2,0) {d};
+
+        \draw (b)--(a);
+        \draw (b)--(f);
+        \draw (b)--(e);
+        \draw (a)--(c);
+        \draw (a)--(d);
+
+        \end{tikzpicture}
+        \\
+        $G \setminus \{\{a,f\},\{c,d\},\{d,e\}\}$
+
+\end{enumerate}
+
+\end{document}
+
diff --git a/school/di-ma/uebung/04/04_3.tex b/school/di-ma/uebung/04/04_3.tex
new file mode 100644
index 0000000..2f7b40a
--- /dev/null
+++ b/school/di-ma/uebung/04/04_3.tex
@@ -0,0 +1,118 @@
+\documentclass[12pt,a4paper,german]{article}
+\usepackage{url}
+%\usepackage{graphics}
+\usepackage{times}
+\usepackage[T1]{fontenc}
+\usepackage{ngerman}
+\usepackage{float}
+\usepackage{diagbox}
+\usepackage[utf8]{inputenc}
+\usepackage{geometry}
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{csquotes}
+\usepackage{graphicx}
+\usepackage{epsfig}
+\usepackage{paralist}
+\usepackage{tikz}
+\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
+
+%%%%%%%%%% Fill out the the definitions %%%%%%%%%
+\def \name {Valentin Brandl}					%
+\def \matrikel {108018274494}				%
+\def \pname {Marvin Herrmann}				%
+\def \pmatrikel {108018265436}				%
+\def \gruppe {2 (Mi 10-12 Andre)}
+\def \qname {Pascal Brackmann}
+\def \qmatrikel {108017113834}				%
+\def \uebung {4}								%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ % DO NOT MODIFY THIS HEADER
+\newcommand{\hwsol}{
+\vspace*{-2cm}
+\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
+\noindent \pmatrikel \quad \pname \\
+\noindent \qmatrikel \quad \qname \\
+\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
+}
+
+\begin{document}
+%Import header
+\hwsol
+
+\section*{Aufgabe 4.3}
+\begin{enumerate}[1.]
+
+    \item Graph $G=(V,E)$ mit\\
+        Knotenmenge: $V=\{1,2,3,4,5,6\}$\\
+        Kantenmenge: $E=
+        \{
+            (1,2),
+            (1,3),
+            (2,3),
+            (3,4),
+            (4,5),
+            (5,6),
+            (6,4)
+        \}$
+        \\
+
+        Adjazenz-Matrix:\\
+        \begin{tabular}{c|cccccc}
+        & 1 &  2 & 3 & 4 & 5 & 6 \\
+        \hline
+            1 & 0 & 1 & 1 & 0 & 0& 0 \\
+            2 & 0 & 0 & 1 & 0 & 0& 0 \\
+            3 & 0 & 0 & 0 & 1 & 0& 0 \\
+            4 & 0 & 0 & 0 & 0 & 1& 0 \\
+            5 & 0 & 0 & 0 & 0 & 0& 1 \\
+            6 & 0 & 0 & 0 & 1 & 0& 0 \\
+        \end{tabular}
+        \\
+        Inzidenz-Matrix:\\
+        \begin{tabular}{c|ccccccc}
+            1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
+            2 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
+            3 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
+            4 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\
+            5 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
+            6 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
+        \end{tabular}
+
+    \item \begin{tikzpicture}
+            \def \n {5}
+            \def \radius {3cm}
+            \def \margin {8} % margin in angles, depends on the radius
+            \node[draw, circle] (1) at (0,1) {1};
+            \node[draw, circle] (4) at (2,2) {4};
+            \node[draw, circle] (6)  at (2,0) {6};
+            \node[draw, circle] (3)  at (1,0) {3};
+            \draw [->] 
+            (1) edge (3)
+            (3) edge (4)
+            (6) edge (4);
+        \end{tikzpicture}
+        \\
+        $V'=\{1,3,4,6\}$
+
+    \item \begin{tikzpicture}
+            \def \n {5}
+            \def \radius {3cm}
+            \def \margin {8} % margin in angles, depends on the radius
+            \node[draw, circle] (1) at (0,0) {1};
+            \node[draw, circle] (3) at (1,0) {3};
+            \node[draw, circle] (4)  at (2,0) {4};
+            \node[draw, circle] (5)  at (3,0) {5};
+            \draw [->] 
+            (1) edge (3)
+            (3) edge (4)
+            (4) edge (5);
+        \end{tikzpicture}
+        \\
+        $V'=\{1,3,4,5\}$
+
+\end{enumerate}
+
+\end{document}
+
diff --git a/school/di-ma/uebung/04/04_4.tex b/school/di-ma/uebung/04/04_4.tex
new file mode 100644
index 0000000..83d4b07
--- /dev/null
+++ b/school/di-ma/uebung/04/04_4.tex
@@ -0,0 +1,77 @@
+\documentclass[12pt,a4paper,german]{article}
+\usepackage{url}
+%\usepackage{graphics}
+\usepackage{times}
+\usepackage[T1]{fontenc}
+\usepackage{ngerman}
+\usepackage{float}
+\usepackage{diagbox}
+\usepackage[utf8]{inputenc}
+\usepackage{geometry}
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{csquotes}
+\usepackage{graphicx}
+\usepackage{epsfig}
+\usepackage{paralist}
+\usepackage{tikz}
+\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
+
+%%%%%%%%%% Fill out the the definitions %%%%%%%%%
+\def \name {Valentin Brandl}					%
+\def \matrikel {108018274494}				%
+\def \pname {Marvin Herrmann}				%
+\def \pmatrikel {108018265436}				%
+\def \gruppe {2 (Mi 10-12 Andre)}
+\def \qname {Pascal Brackmann}
+\def \qmatrikel {108017113834}				%
+\def \uebung {4}								%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ % DO NOT MODIFY THIS HEADER
+\newcommand{\hwsol}{
+\vspace*{-2cm}
+\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
+\noindent \pmatrikel \quad \pname \\
+\noindent \qmatrikel \quad \qname \\
+\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
+}
+
+\begin{document}
+%Import header
+\hwsol
+
+\section*{Aufgabe 4.4}
+
+Zu zeigen: Wenn ein Baum genau $k \geq 1$ Knoten vom Grad 4 enthält (außer Blätter), dann besitzt der Baum mindestens
+$2 \cdot k + 2$ Blätter.\\
+\\
+IA: Ein Baum mit nur einem Knoten von Grad 4 muss 4 Blätter haben
+
+\begin{tikzpicture}
+    \node[draw,circle] (1) at (2,1) {1};
+    \node[draw,circle] (2) at (0,0) {2};
+    \node[draw,circle] (3) at (1,0) {3};
+    \node[draw,circle] (4) at (3,0) {4};
+    \node[draw,circle] (5) at (4,0) {5};
+
+    \draw (1)--(2);
+    \draw (1)--(3);
+    \draw (1)--(4);
+    \draw (1)--(5);
+\end{tikzpicture}
+\\
+$A(1) = 2 \cdot 1 + 2 = 4$\\
+\\
+\textbf{IV: $A(k) = 2 \cdot k +2$}\\
+\\
+IS: $k \rightarrow k+1$\\
+Um einen Knoten mit Grad 4 hinzuzufügen, kann man nun eines der Blätter nehmen und drei Blätter anhängen. Man bekommt
+also 3 Blätter hinzu, verliert aber auch eines, da dieses zum neuen Knoten wird.\\
+\\
+$A(k+1) = A(k) + 3 - 1 \overset{(IV)}{=} (2 \cdot k+2)+3-1$ 
+$= 2 \cdot (k+1)+2$\\
+q.e.d
+
+\end{document}
+
diff --git a/school/di-ma/uebung/04/pascal_04.tex b/school/di-ma/uebung/04/pascal_04.tex
new file mode 100644
index 0000000..1c4aa9a
--- /dev/null
+++ b/school/di-ma/uebung/04/pascal_04.tex
@@ -0,0 +1,246 @@
+\documentclass[10pt,a4paper]{article}
+\usepackage[utf8x]{inputenc}
+\usepackage{ucs}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{graphicx}
+\usepackage{tikz}
+\title{Blatt 04}
+\begin{document}
+
+\maketitle
+\newpage
+\section{Aufgabe 1}
+TODO
+\section{Aufgabe 2}
+
+\begin{tikzpicture}
+\def \n {5}
+\def \radius {3cm}
+\def \margin {8} % margin in angles, depends on the radius
+\node[draw, circle] (a) at (0,1) {a};
+\node[draw, circle] (b)  at (1,2) {b};
+\node[draw, circle] (c) at (2,2) {c};
+\node[draw, circle] (d)  at (3,1) {d};
+\node[draw, circle] (e)  at (2,0) {e};
+\node[draw, circle] (f)  at (1,0) {f};
+\draw (a)--(b);
+\draw (a)--(c);
+\draw (a)--(d);
+\draw (a)--(f);
+\draw (b)--(f);
+\draw (b)--(e);
+\draw (c)--(d);
+\draw (d)--(e);
+\end{tikzpicture}
+\\
+\subsection{1.)}
+Graph $G=(V,E)$ mit\\
+Knotenmenge: $V=\{a,b,c,d,e,f\}$\\
+Kantenmenge: $E=
+\{
+\{a,b\},
+\{a,c\},
+\{a,d\},
+\{a,f\},
+\{b,f\},
+\{b,e\},
+\{c,d\},
+\{d,e\}
+\}
+$
+\\
+Adjazens-Matrix:\\
+\begin{tabular}{c|ccccccc}
+& a & b & c & d & e &  f \\
+\hline
+a & 0 & 1 & 1 & 1 & 0 & 1 \\
+b & 1 & 0 & 0 & 0 & 1 & 1 \\
+c & 1 & 0 & 0 & 1 & 0 & 0 \\
+d & 1 & 0 & 1 & 0 & 1 & 0 \\
+e & 0 & 1 & 0 & 1 & 0 & 0 \\
+f & 1 & 1 & 0 & 0 & 0 & 0 \\
+\end{tabular} 
+\\
+\\
+Inz.-Matrix:\\
+\begin{tabular}{c|ccccccccc}
+a & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
+b & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
+c & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\
+d & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\
+e & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\
+f & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
+\end{tabular} 
+
+\subsection{2.)}
+\begin{tikzpicture}
+\def \n {5}
+\def \radius {3cm}
+\def \margin {8} % margin in angles, depends on the radius
+\node[draw, circle] (a) at (3,2) {a};
+\node[draw, circle] (b)  at (1,1) {b};
+\node[draw, circle] (c) at (2,1) {c};
+\node[draw, circle] (d)  at (3,1) {d};
+\node[draw, circle] (f)  at (4,1) {f};
+\node[draw, circle] (e)  at (0,0) {e};
+
+\draw (a)--(b);
+\draw (a)--(c);
+\draw (a)--(d);
+\draw (a)--(f);
+\draw (b)--(e);
+\end{tikzpicture}
+\\
+$G \setminus \{\{b,f\},\{c,d\},\{d,e\}\}$
+
+\begin{tikzpicture}
+\def \n {5}
+\def \radius {3cm}
+\def \margin {8} % margin in angles, depends on the radius
+\node[draw, circle] (b) at (2,2) {b};
+\node[draw, circle] (a)  at (1,1) {a};
+\node[draw, circle] (f) at (2,1) {f};
+\node[draw, circle] (e)  at (3,1) {e};
+\node[draw, circle] (c)  at (0,0) {c};
+\node[draw, circle] (d)  at (2,0) {d};
+
+\draw (b)--(a);
+\draw (b)--(f);
+\draw (b)--(e);
+\draw (a)--(c);
+\draw (a)--(d);
+
+\end{tikzpicture}
+\\
+$G \setminus \{\{a,f\},\{c,d\},\{d,e\}\}$
+
+
+
+
+
+
+\section{Aufgabe 3}
+
+
+\begin{tikzpicture}
+\def \n {5}
+\def \radius {3cm}
+\def \margin {8} % margin in angles, depends on the radius
+\node[draw, circle] (1) at (0,1) {1};
+\node[draw, circle] (2)  at (1,2) {2};
+\node[draw, circle] (4) at (2,2) {4};
+\node[draw, circle] (5)  at (3,1) {5};
+\node[draw, circle] (6)  at (2,0) {6};
+\node[draw, circle] (3)  at (1,0) {3};
+
+\draw [->] 
+(1) edge (2)
+(1) edge (3)
+(2) edge (3)
+(3) edge (4)
+(4) edge (5)
+(5) edge (6)
+(6) edge (4);
+\end{tikzpicture}
+
+\subsection{1.)}
+Graph $G=(V,E)$ mit\\
+Knotenmenge: $V=\{1,2,3,4,5,6\}$\\
+Kantenmenge: $E=
+\{
+(1,2),
+(1,3),
+(2,3),
+(3,4),
+(4,5),
+(5,6),
+(6,4)
+\}$
+\\
+
+Adjazens-Matrix:\\
+\begin{tabular}{c|cccccc}
+& 1 &  2 & 3 & 4 & 5 & 6 \\
+\hline
+1 & 0 & 1 & 1 & 0 & 0& 0 \\
+2 & 0 & 0 & 1 & 0 & 0& 0 \\
+3 & 0 & 0 & 0 & 1 & 0& 0 \\
+4 & 0 & 0 & 0 & 0 & 1& 0 \\
+5 & 0 & 0 & 0 & 0 & 0& 1 \\
+6 & 0 & 0 & 0 & 1 & 0& 0 \\
+\end{tabular}
+\\
+Inzid.-Matrix:\\
+\begin{tabular}{c|ccccccc}
+1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
+2 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
+3 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
+4 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\
+5 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
+6 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
+\end{tabular}
+\subsection{2.)}
+\begin{tikzpicture}
+\def \n {5}
+\def \radius {3cm}
+\def \margin {8} % margin in angles, depends on the radius
+\node[draw, circle] (1) at (0,1) {1};
+\node[draw, circle] (4) at (2,2) {4};
+\node[draw, circle] (6)  at (2,0) {6};
+\node[draw, circle] (3)  at (1,0) {3};
+\draw [->] 
+(1) edge (3)
+(3) edge (4)
+(6) edge (4);
+\end{tikzpicture}
+\\
+$V'=\{1,3,4,6\}$\\
+\\
+\subsection{3.)}
+\begin{tikzpicture}
+\def \n {5}
+\def \radius {3cm}
+\def \margin {8} % margin in angles, depends on the radius
+\node[draw, circle] (1) at (0,0) {1};
+\node[draw, circle] (3) at (1,0) {3};
+\node[draw, circle] (4)  at (2,0) {4};
+\node[draw, circle] (5)  at (3,0) {5};
+\draw [->] 
+(1) edge (3)
+(3) edge (4)
+(4) edge (5);
+\end{tikzpicture}
+\\
+$V'=\{1,3,4,5\}$\\
+\section{Aufgabe 4}
+
+Zu zeigen: Wenn ein Baum genau $k \geq 1$ Knoten vom Grad 4 enthält (außer Blätter), dann besitzt der Baum mindestenz $2 \cdot k + 2$ Blätter.\\
+\\
+IA: Ein Baum mit nur einem Knoten von Grad 4 muss 4 Blätter haben
+
+\begin{tikzpicture}
+\node[draw,circle] (1) at (2,1) {1};
+\node[draw,circle] (2) at (0,0) {2};
+\node[draw,circle] (3) at (1,0) {3};
+\node[draw,circle] (4) at (3,0) {4};
+\node[draw,circle] (5) at (4,0) {5};
+
+\draw (1)--(2);
+\draw (1)--(3);
+\draw (1)--(4);
+\draw (1)--(5);
+\end{tikzpicture}
+\\
+$A(1) = 2 \cdot 1 + 2 = 4$\\
+\\
+\textbf{IV: $A(k) = 2 \cdot k +2$}\\
+\\
+IS: $k \rightarrow k+1$\\
+Um einen Knoten mit Grad 4 hinzuzufügen, kann man nun eines der Blätter nehmen und drei Blätter anhängen. Man bekommt also 3 Blätter hinzu, verliert aber auch eines, da dieses zum neuen Knoten wird.\\
+\\
+$A(k+1) = A(k) + 3 - 1 \overset{(IV)}{=} (2 \cdot k+2)+3-1$ 
+$= 2 \cdot (k+1)+2$\\
+q.e.d
+\end{document}
\ No newline at end of file