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\title{General Mathematics and CPS II}
\author{Exercise 3}
\date{February 10, 2012}
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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 A standard deck of cards is dealt into 13 piles of 4 cards each.
Show that you can always choose exactly one card from each of the 13
piles as to obtain exactly one card of each rank, i.e., each of 2, 3,
4, \ldots, King, Ace.

\item Ivanov, p.\ 99, Problem 13.

(This problem will be discussed in class on Wednesday again, so
you may choose to hand it in on Friday.)
\end{enumerate}
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