**UPDATE (2011/09/20):** This post was superseded by An improved proof of the handshaking lemma.

In graph theory, the degree of a vertex $A$, $\fapp{d}{A}$, is the number of edges incident with the vertex $A$, counting loops twice. So, considering Graph 0 below, we have $\fapp{d}{A}=3$, $\fapp{d}{B}=3$, $\fapp{d}{C}=1$, $\fapp{d}{D}=3$, and $\fapp{d}{E}=2$.

Graph 0: Example of an undirected graph with five nodes

A well-known property is that every undirected graph contains an even number of vertices with odd degree. The result first appeared in Euler’s 1736 paper on the Seven Bridges of Königsberg and is also known as the handshaking lemma (that’s because another way of formulating the property is that the number of people that have shaken hands an odd number of times is even).

As we can easily verify, Graph 0 satisfies this property. There are four vertices with odd degree ($A$,$B$, $C$, and $D$), and 4, of course, is an even number.

Although the proof of this property is simple, I have never seen it proved in a calculational and goal-oriented way. My aim with this post is to show you a development of a goal-oriented proof.

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