Euler graph and Eulerian path

graph theory

is it always true that if a graph contains a Eulerian path(at least one) then it will always be a Euler graph?

I know that all the vertices of a graph should be of even degree to become a Euler graph as well as the graph should be connected too, but was having this self-doubt.

EDIT:
what about this graph:

enter image description here

Euler or not ?

Best Answer

According to Wolfram Mathworld an Euler graph is a graph containing an Eulerian cycle. There surely are examples of graphs with an Eulerian path, but not an Eulerian cycle. Consider two connected vertices for example.

EDIT: The link also mentions some authors define an Euler graph as a connected graph where every vertex has even degree. It is by no means trivial to show equivalence between these definition, but one of the two implications is pretty easy. If a graph has a Eulerian cycle, then every vertex must be entered and left an equal amount of times in the cycle. Since every edge can only be visited once, we find an even amount of edges per vertex. ($2$ times the amount of times the vertex is visited in the cycle)

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