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I am trying to prove Let G be a connected graph with one edge such that every vertex has even positive degree. Prove that G has an Euler circuit.

I know that a graph is an Euler circuit iff it is connected and degree of each vertex is even. How it is possible that one edge can make even degree?

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  • $\begingroup$ That's probably a typo $\endgroup$
    – Yanko
    Commented Feb 1, 2023 at 3:09

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A graph with one vertex and no edges has all vertices of even degree. This is an edge case for the existence of a Eulerian circuit. If your definition does not allow this graph to have an Eulerian circuit the requirement of one edge is needed. The empty path uses all the edges, but whether it is a circuit is difficult.

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