71 lines
2.1 KiB
TeX
71 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{cancel}
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\usepackage{wasysym}
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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 {9} %
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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 9.2}
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\begin{enumerate}[1.]
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\item Ein $K_n$ mit $n \equiv 0 \mod 2$ ($n$ ist gerade) hat 2 perfekte Matchings, die jeweils versetzt jede zweite
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Kante des Kreises beinhalten
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Für ein perfektes Matching muss gelten $|M| = \frac{|V|}{2}$. Für einen Kreis $K_n$ mit $n \equiv 1 \mod 2$ ($n$
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ist ungerade) gilt $|V| = n$. Da $n$ ungerade existiert keine ganze Zahl $\frac{n}{2}$ $\Rightarrow$ es
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existiert kein perfektes Matching.
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\item
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\begin{enumerate}[i)]
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\item $M_1 = \{ \{000,100\}, \{001,101\}, \{010,110\}, \{011,111\} \}$
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\item $M_2 = \{ \{000,001\}, \{100,101\}, \{010,011\}, \{110,111\} \}$
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\item $M_3 = \{ \{000,010\}, \{001,011\}, \{100,110\}, \{101,111\} \}$
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\end{enumerate}
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Es gibt 3 perfekte Matchings.
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\item $\{ \{000,100\}, \{001,101\}, \{110,111\}, \{010,011\} \}$
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\end{enumerate}
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\end{document}
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