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\begin{document}
%{\sf KTH Matematik}
\begin{center} 
 {\bf 5B1118 Diskret Matematik \\
Kontrollskrivning 3\\
Fredagen  den  2 April, 2004 \\}
\end{center}
%\vspace{.1in}
\begin{itemize}
\item {\it Skrivtid:}{\bf 9:15-10:15.}
\item {\it Till{\aa}tna hj\"alpmedel: Minir\"aknare med sifferdisplay.}
\item {\it\underline{Motivering kr\"avs}!}
\item {\it F\"or godk\"ant resultat kr\"avs minst 5 po\"ang.}
\end{itemize}
\noindent\framebox[15cm]{Personnummer:\hspace{12cm}}\\
\framebox[15cm]{Namn:\hspace{12cm}}\\
%\vspace{.1in}
\begin{enumerate}
\item (3p) L\aa t $G=(V,E)$ vara en enkel graf.
\begin{itemize}
\item Best\"am $|V|$ om $|E|=9$ och varje nod har grad $3$.
\item Ge en geometrisk representation av $G$, dvs rita grafen $G$.
\item \"Ar $G$ en eulersk graf?(Motivera ditt svar)
\end{itemize}

\vspace{.3in}

\item(3p) Avg\"or om dessa grafer \"ar isomorfa.(Motivera ditt svar).

 $G_1=$\setlength{\unitlength}{1cm}
\begin{picture}(3,2.2) 
\put(2,0){\circle*{0.1}}
\put(0,2){\circle*{0.1}}
\put(0,0){\circle*{0.1}}
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\put(0,0){\line(1,1){2}}
\put(2,0){\line(0,1){2}}
\put(2,0){\line(-1,1){2}}
\put(0,2){\line(1,0){2}}
\end{picture}
$G_2$= \begin{picture}(3,2.2)
\put(2,0){\circle*{0.1}}
\put(0,2){\circle*{0.1}}
\put(0,0){\circle*{0.1}}
\put(2,2){\circle*{0.1}}
\put(3,1){\circle*{0.1}}
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\put(2,2){\line(1,-1){1}}
\end{picture}

\vspace{.2in}

\item(3p) \begin{itemize} \item Rita en graf med incidensmatris:
\[ A_G=\left( \begin{array}{ccccc}
1&0&0&0&0\\
0&1&1&0&0\\
0&0&1&1&0\\
0&0&0&1&1\\
1&1&0&0&1
\end{array} \right)\]
\item \"Ar $G$ ett tr\"ad? (Motivera ditt svaret.)
\item Best\"am $\chi(G)$.
\end{itemize}
\end{enumerate}

\vspace{1in}


betyg:
\begin{tabular}{|c|c|c|c|}\hline
Upp.(1)& Upp. (2)&Upp. (3)&Tot.\\ \hline
&&&\\ \hline
\end{tabular}
\newpage 
KTH, Matematik, 5B1118, KS3.\hspace{1in}Namn: 
\newpage 
KTH, Matematik, 5B1118, KS3.\hspace{1in}Namn: 
\newpage 
KTH, Matematik, 5B1118, KS3.\hspace{1in}Namn:
\end{document}