78 lines
2.1 KiB
TeX
78 lines
2.1 KiB
TeX
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\documentclass[12pt,a4paper,german]{article}
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\usepackage{url}
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%\usepackage{graphics}
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\usepackage{times}
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\usepackage[T1]{fontenc}
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\usepackage{ngerman}
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\usepackage{float}
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\usepackage{diagbox}
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\usepackage[utf8]{inputenc}
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\usepackage{geometry}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{epsfig}
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\usepackage{paralist}
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\usepackage{tikz}
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\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
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\def \name {Valentin Brandl} %
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\def \matrikel {108018274494} %
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\def \pname {Marvin Herrmann} %
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\def \pmatrikel {108018265436} %
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\def \gruppe {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {4} %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DO NOT MODIFY THIS HEADER
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\newcommand{\hwsol}{
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\vspace*{-2cm}
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\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
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\noindent \pmatrikel \quad \pname \\
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\noindent \qmatrikel \quad \qname \\
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\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
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}
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\begin{document}
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%Import header
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\hwsol
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\section*{Aufgabe 4.4}
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Zu zeigen: Wenn ein Baum genau $k \geq 1$ Knoten vom Grad 4 enthält (außer Blätter), dann besitzt der Baum mindestens
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$2 \cdot k + 2$ Blätter.\\
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\\
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IA: Ein Baum mit nur einem Knoten von Grad 4 muss 4 Blätter haben
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\begin{tikzpicture}
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\node[draw,circle] (1) at (2,1) {1};
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\node[draw,circle] (2) at (0,0) {2};
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\node[draw,circle] (3) at (1,0) {3};
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\node[draw,circle] (4) at (3,0) {4};
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\node[draw,circle] (5) at (4,0) {5};
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\draw (1)--(2);
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\draw (1)--(3);
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\draw (1)--(4);
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\draw (1)--(5);
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\end{tikzpicture}
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\\
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$A(1) = 2 \cdot 1 + 2 = 4$\\
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\\
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\textbf{IV: $A(k) = 2 \cdot k +2$}\\
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\\
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IS: $k \rightarrow k+1$\\
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Um einen Knoten mit Grad 4 hinzuzufügen, kann man nun eines der Blätter nehmen und drei Blätter anhängen. Man bekommt
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also 3 Blätter hinzu, verliert aber auch eines, da dieses zum neuen Knoten wird.\\
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\\
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$A(k+1) = A(k) + 3 - 1 \overset{(IV)}{=} (2 \cdot k+2)+3-1$
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$= 2 \cdot (k+1)+2$\\
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q.e.d
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\end{document}
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