Example 1: Finding an Euler circuit

All of the vertices have even degree except B (degree 3) and D (degree 3). According to Euler's Theorems, there is no Euler circuit for this graph. (However, there is an Euler path, starting at one of the odd vertices and ending at the other).

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Example 1: Finding a Hamilton circuit

One possible solution: AGFECDBA

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Example 2: Finding an Euler circuit

Since there are no odd vertices, there is an Euler circuit.

One possible solution: AGFADCFECBA

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Example 2: Finding a Hamilton circuit

There is no Hamilton circuit.
Think of it this way. Any circuit can be written so that it starts at any vertex you like. So suppose you try to follow a circuit starting at vertex B. No matter whether you travel left or right, there is no way for you to go back to B without retracing an edge unless you skip vertex D. But if you skip vertex D, then you don't have a Hamilton circuit.

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