From 4d1a6e3e3252f521bd6ec27588a04925aab7d894 Mon Sep 17 00:00:00 2001 From: Valentin Brandl Date: Wed, 28 Nov 2018 19:02:38 +0100 Subject: [PATCH] Add solution for dima 7.2 and 7.4 --- school/di-ma/uebung/07/07_2.tex | 101 +++++++++++++++++++++++++++++ school/di-ma/uebung/07/07_2_1.dot | 12 ++++ school/di-ma/uebung/07/07_2_2.dot | 17 +++++ school/di-ma/uebung/07/07_2_3.dot | 16 +++++ school/di-ma/uebung/07/07_2_4.dot | 5 ++ school/di-ma/uebung/07/07_4.tex | 85 ++++++++++++++++++++++++ school/di-ma/uebung/07/07_4_1.jpg | Bin 0 -> 3380204 bytes school/di-ma/uebung/07/07_4_2a.jpg | Bin 0 -> 3826415 bytes school/di-ma/uebung/07/07_4_2b.jpg | Bin 0 -> 3608198 bytes 9 files changed, 236 insertions(+) create mode 100644 school/di-ma/uebung/07/07_2.tex create mode 100644 school/di-ma/uebung/07/07_2_1.dot create mode 100644 school/di-ma/uebung/07/07_2_2.dot create mode 100644 school/di-ma/uebung/07/07_2_3.dot create mode 100644 school/di-ma/uebung/07/07_2_4.dot create mode 100644 school/di-ma/uebung/07/07_4.tex create mode 100644 school/di-ma/uebung/07/07_4_1.jpg create mode 100644 school/di-ma/uebung/07/07_4_2a.jpg create mode 100644 school/di-ma/uebung/07/07_4_2b.jpg diff --git a/school/di-ma/uebung/07/07_2.tex b/school/di-ma/uebung/07/07_2.tex new file mode 100644 index 0000000..d054920 --- /dev/null +++ b/school/di-ma/uebung/07/07_2.tex @@ -0,0 +1,101 @@ +\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{cancel} +\usepackage{wasysym} +\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 {7} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % 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 7.2} + +\begin{enumerate}[1.] + + \item \includegraphics[scale=0.5]{./build/school/di-ma/uebung/07/07_2_1.png} + + \begin{itemize} + \item Eulertour: $(1, 2, 3, 1)$ + \item Hamiltonkreis $(1, 2, 3, 1)$ + \end{itemize} + + \item \includegraphics[scale=0.5]{./build/school/di-ma/uebung/07/07_2_2.png} + + \begin{itemize} + \item Eulertour: $(1,2,3,4,5,3,1)$ + \item Kein Hamiltonkreis: (Widerspruchsbeweis) + + G sei hamiltonsch. Aufgrund der Symmetrie gibt es nur 2 Mögliche Startknoten für einen Hamiltonkreis: + Der mittlere Knoten ($3$) oder einer der 4 äußeren Knoten ($1,2,4,5$). + + \begin{enumerate}[1.] + \item Fall: Starten bei Knoten $3$ + + Aufgrund der Symmetrie sind alle möglichen Wege von hier aus äquivalent. Nach $(3,1,2)$ ist es + unmöglich, die restlichen Knoten des Graphen zu besuchen, ohne $3$ erneut zu besuchen. + + $\Rightarrow$ Kein Hamiltonkreis \lightning + + \item Fall: Starten bei Knoten $1$ (oder äquivalent $2,4,5$) + + Der einzige Weg, um die gegenüberliegenden Knoten ($4,5$) zu erreichen, führt über $3$. Um einen + Kreis zu bilden, muss man aber wieder zurück zu $1$ laufen. Dieser Weg führt erneut über $3$, + $3$ wird also doppelt durchlaufen + + $\Rightarrow$ Kein Hamiltonkreis \lightning + \end{enumerate} + + $\Rightarrow$ G ist nicht hamiltonsch + + \end{itemize} + + \item \includegraphics[scale=0.5]{./build/school/di-ma/uebung/07/07_2_3.png} + + \begin{itemize} + \item Keine Eulertour, da $deg(1) = deg(4) = 3$ ist nicht gerade + \item Hamiltonkreis $(1, 2, 3, 4, 1)$ + \end{itemize} + + \item \includegraphics[scale=0.5]{./build/school/di-ma/uebung/07/07_2_4.png} + + Graph ist nicht zusammenhängend, daher ist die Grundvoraussetzung für \enquote{eulersch} und + \enquote{hamiltonsch} nicht gegeben $\Rightarrow$ nicht eulersch und nicht hamiltonsch + +\end{enumerate} + +\end{document} diff --git a/school/di-ma/uebung/07/07_2_1.dot b/school/di-ma/uebung/07/07_2_1.dot new file mode 100644 index 0000000..886b72e --- /dev/null +++ b/school/di-ma/uebung/07/07_2_1.dot @@ -0,0 +1,12 @@ +graph { + node [ shape="circle" ] + { + rank=same; + 1; + } + { + rank=same; + 2; 3; + } + 1 -- 2 -- 3 -- 1; +} diff --git a/school/di-ma/uebung/07/07_2_2.dot b/school/di-ma/uebung/07/07_2_2.dot new file mode 100644 index 0000000..c37dd8e --- /dev/null +++ b/school/di-ma/uebung/07/07_2_2.dot @@ -0,0 +1,17 @@ +graph { + node [ shape="circle" ] + { + rank=same; + 1; 2; + } + { + rank=same; + 3; + } + { + rank=same; + 4; 5; + } + 1 -- 2 -- 3 -- 1; + 3 -- 4 -- 5 -- 3; +} diff --git a/school/di-ma/uebung/07/07_2_3.dot b/school/di-ma/uebung/07/07_2_3.dot new file mode 100644 index 0000000..f2db934 --- /dev/null +++ b/school/di-ma/uebung/07/07_2_3.dot @@ -0,0 +1,16 @@ +graph { + node [ shape="circle" ] + { + rank=same; + 1; 2; + 1 -- 2; + } + { + rank=same; + 3; 4; + 3 -- 4; + } + 1 -- 3; + 2 -- 4; + 1 -- 4; +} diff --git a/school/di-ma/uebung/07/07_2_4.dot b/school/di-ma/uebung/07/07_2_4.dot new file mode 100644 index 0000000..a3fab8b --- /dev/null +++ b/school/di-ma/uebung/07/07_2_4.dot @@ -0,0 +1,5 @@ +graph { + node [ shape="circle" ] + 1; + 2; +} diff --git a/school/di-ma/uebung/07/07_4.tex b/school/di-ma/uebung/07/07_4.tex new file mode 100644 index 0000000..63b4339 --- /dev/null +++ b/school/di-ma/uebung/07/07_4.tex @@ -0,0 +1,85 @@ +\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{cancel} +\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 {7} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % 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 7.4} + +\begin{enumerate}[1.] + + \item Sei $G = (V, E)$ ein Graph mit $|V| = 7$ und $|E| = 16$ + + Ein Graph ist planar, falls $|E| \leq 3 |V| - 6$ + + $16 \geq 3 * 7 - 6 = 15 \Rightarrow G$ ist nicht planar + + \begin{figure}[h!] + \centering + \includegraphics[width=\textwidth]{./school/di-ma/uebung/07/07_4_1.jpg} + \caption{Planarer Graph $G=(V,E)$ mit $|V| = 7$ und $|E| = 15$} + \end{figure} + + \item Ein Teilgraph von $G$ lässt sich folgendermaßen Zeichnen (Abbildung \ref{TG}): + + \begin{figure}[h!] + \centering + \includegraphics[width=\textwidth/2]{./school/di-ma/uebung/07/07_4_2a.jpg} + \caption{Teilgraph von G} + \label{TG} + \end{figure} + + Nach entfernen der Unterteilungen durch die Konten $3,6,7$ und $9$, erhält man folgenden Graphen (Abbildung + \ref{TGU}): + + \begin{figure}[h!] + \centering + \includegraphics[width=\textwidth/2]{./school/di-ma/uebung/07/07_4_2b.jpg} + \caption{Teilgraph von G nach entfernen der Unterteilungen} + \label{TGU} + \end{figure} + + Dieser Graph ist $K_5$. Damit wurde gezeigt, dass $G$ den Graphen $K_5$ enthält. $K_5$ ist nicht planar und + damit ist auch jeder Graph, der $K_5$ enthält, nicht planar. + +\end{enumerate} + +\end{document} diff --git a/school/di-ma/uebung/07/07_4_1.jpg b/school/di-ma/uebung/07/07_4_1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..01b384aa361c04420411f2a18b36738a33130362 GIT binary patch literal 3380204 zcmeFZc{o(>`#65aV2Fq)3L|4r#UQ&O#ZbsvBxFgJ?E5yB$`Z!dqO92!+JwqhGDygt zH4Vm)B{Q0aF@C3B@AvET{r&g*{{H)3*Y{lGoadbDex7sg=ehU$e(X=|&&z9E2z9>+ z0LI3EEC2v(0BfQIzznt^;O`;RVF3EK{pU{lKW!$keH>u<2L`ZCaQ`nna1(Ou-{`(__vMDVEVUxxtQs{Z#txt0RayD zGcx3AI`rRh9B~i;20mwoFhPN0=zrlk;1PJ$M6~GmO5P(B}pEDW%?eg~o!T^x}dGDWRJbxa6e**ux9{GD^{h#GzGGzL9 zl21)CLH;{AOg0ece*kpJ7qemlU!25drfpb#;Di3bAZf$Voe9)rlRK>i;8d^>p0 z1Z8Gng|V?8-~i_o0);X$Ls?juL7sp_{h=ym9v0pc3c9R^&2GTN{P>WuxDqzRx#u1H z=3h3&6`lO!*bf{zDj;}FLQ+avMpjArtct3dy59K<`UYr2BMVC_>nm4nY@JTlW($X@@T1D^lE=zqb72gHYonHkCq 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