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\begin{definition}[ Face degrees ]
Let $G=(V,E)$ be a connected multigraph embedded in the plane
and let $f$ be a face of this embedding.
We define the {\bf degree} of $f$, denoted $\deg(f)$, to be the number
of edges in a facial walk of $f$.
\end{definition}

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Example
\vspace{3cm}
\begin{theorem}
If $G$ has faces $f_1,f_2, \dots,f_k$ then \ $\ds \sum_{i=1}^{k} \deg(f_i) = 2|E|$.
\end{theorem}
Proof

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