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\homework{4: Induction, the Euler Characteristic, and Chemistry}{Due Tuesday, Week 5, at the start of class}
Work on these problems for 90m/until you solve everything/longer if you're interested!
\begin{enumerate}

\item We can define a \textbf{graph-fullerene} as any graph $G$ with the following properties:
\begin{itemize}
\item Every vertex in our graph is connected to three other vertices.
\item Our graph is planar.
\item Every face of our graph is either a pentagon or a hexagon.
\end{itemize}
Prove the claim we made in class:\ every graph-fullerene contains exactly twelve pentagons.  (Use the Euler characteristic!)
\end{enumerate}
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