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\title{General Mathematics and CPS II}
\author{Exercise 3}
\date{February 14, 2014}
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\begin{enumerate}

\item Let $G$ be a finite connected planar graph with $V$ its set of
vertices, $E$ its set of edges, and $F$ its set of faces.
\begin{enumerate}
\item Show that $2 \, \lvert E \rvert \geq 3 \, \lvert F \rvert$.
\item Show that $\lvert E \rvert \leq 3 \, \lvert V \rvert - 6$.
\item Conclude that every planar graph must have at least one vertex
of valency less than $6$.
\end{enumerate}

\item Write out the proof for Ivanov, p.\ 96, Lemma 5.

\end{enumerate}
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