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\begin{document}
\noindent{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1516}{2015-16}:
  \hfill\url{http://drorbn.net/?title=15-344}
}

\begin{center}

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\noindent{\bf Do not turn this page until instructed.}

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\href{http://drorbn.net/?title=15-344}{\large Math 344 Combinatorics}

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{\LARGE Term Test}

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University of Toronto, October 27, 2015

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\noindent{\bf Solve 4 of the 5 problems on the other side of this page. }\\
Each problem is worth 25 points.\\You have an hour and fifty minutes to
write this test.

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\noindent{\bf Notes}
\begin{itemize}
\item No outside material other than stationary is allowed.
\item {\bf Neatness counts! Language counts!} The {\em ideal} written
solution to a problem looks like a page from a textbook; neat and clean
and made of complete and grammatical sentences. Definitely phrases like
``there exists'' or ``for every'' cannot be skipped. Lectures are mostly
made of spoken words, and so the blackboard part of proofs given
during lectures often omits or shortens key phrases. The ideal written
solution to a problem does not do that.
\end{itemize}

\end{center}

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\begin{center}

\includegraphics[width=3in]{4DCube.png}
\linebreak
{a 4D cube}

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{\bf Good Luck!}

\end{center}

\newpage

\parpic[r]{\includegraphics[scale=0.75]{IsomorphicQ.pdf}}
\noindent{\bf Solve 4 of the following 5 problems. } Each problem is worth
25 points. You have an hour and fifty minutes. {\bf Neatness counts! Language counts!}

\vskip 5mm\noindent{\bf Problem 1. } Consider the two graphs $G_1$ and $G_2$ shown on the right.
\begin{enumerate}
\item For each of these graph decide whether or not it is bipartite.
\item Is $G_1$ isomorphic to $G_2$?
\end{enumerate}

\par\noindent{\small {\bf Tip. } In math exams, yes/no answers must be accompanied with a proof.}

\noindent{\small {\bf Tip. } Don't start working! Read the whole exam first. You may wish to start with the questions that are easiest for you.}

\vskip 5mm
\parpic[r]{\includegraphics[scale=0.9]{MountainRange.pdf}}
\noindent{\bf Problem 2. } Build the range graph for the mountain range shown on the right and use it to solve the ``Mountain Climbers Puzzle'' for that range.

\vskip 5mm\noindent{\bf Problem 3. } Consider the complete bipartite graph $K_{3,3}$.
\begin{enumerate}
\item Prove that it is not planar.
\item Does it have an Euler cycle or path? Justify your answer, and if you are using any theorems proven in class, state them in full.
\end{enumerate}

\vskip 5mm\noindent{\bf Problem 4. } Without using the four colour theorem, prove that if $G$ is a planar graph such that every proper subgraph of $G$ has a 4-colouring and such that $G$ has a vertex of degree $4$, then $G$ has a 4-colouring.

\par\noindent{\small {\bf Tip. } $X'$ is called ``a  proper subset of a set $X$'' if $X'\subset X$ and $X'\neq X$. Likewise, $G'$ is a proper subgraph of a graph $G$ if $G'\subset G$ and $G'\neq G$.}

\vskip 5mm
\parpic[r]{\includegraphics[scale=1]{MexicanHat.pdf}}
\noindent{\bf Problem 5. } Compute the chromatic polynomial of the ``Mexican Hat'' graph shown on the right.

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\centerline{\bf Good Luck!}

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\end{document}