# Can we consider an Euler Circuit as a Euler Path?

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.

• Yes. An Eulerian circuit is a special kind of Eulerian path. – Rob Arthan Jul 30 '18 at 21:04
• @RobArthan That is not true. A circuit is not a path. You have to remove an edge to get a path. – Morgan Rodgers Sep 20 '18 at 17:24
• @MorganRodgers: opinions seem to vary on this point in graph theory. en.wikipedia.org/wiki/Path_(graph_theory) allows a path to start and end at the same vertex (but, oddly, in my opinion, doesn't allow any other repeats in the vertices). I had never noticed (until you raised your comment) that graph theorists have bizarre prejudices about this terminology that algebraic topologists don't. – Rob Arthan Sep 20 '18 at 19:57
• @RobArthan A walk is the term when vertices are possible repeated. But interesting, I did not realize there was debate about the first/last being allowed as identical. – Morgan Rodgers Sep 20 '18 at 20:08

• 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.