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If we have a Graph with Euler Circuit can we the consider it as a special Euler Path that start and end in the same Node?

I am asking because the Condition of Euler Path is that we have 0 or 2 Nodes with an odd degree so but the graph with 0 nodes with odd degrees will have an Euler Circuit.

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1 Answer

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If you take 10 graph theorists then you will have about 50 different definitions of paths and cycles between them.

You should be aware that:

• If you have a connected graph with exactly $2$ vertices of odd degree, then you can start at one and end at the other, using each edge exactly once, but possibly repeating vertices. This is the only degree condition under which this happens.
• If you have a connected graph with exactly $0$ vertices of odd degree, then you can start at any vertex and return to that same vertex, using each edge exactly once, but possibly repeating vertices. This is the only degree condition under which this happens.

If you know this, it doesn't matter if you call these Euler paths, Euler circuits, Euler trails, Euler walks, or Euler meandering throughways.

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