Can an euler circuit contain euler path?

Question

I am just confused but I believe that by definition you a graph cant be both Euler circuit and Euler path. Euler circuit: a graph with all vertices having even degrees. Euler path: a graph with vertices of two odd degrees

Answer 1

Maybe your definitions are a little mixed up?

Euler path - Path that uses each edge exactly once.

Euler circuit - Circuit that uses each edge exactly once.

An Euler circuit starts and ends with the same vertex while an Euler path must start and end with different vertices.

I think you're talking about the equivalences:

$G$ has an Euler path $\iff$ $G$ has two exactly two vertices of odd degree.

$G$ has an Euler circuit $\iff$ each vertex in $G$ is of even degree.

So the two are naturally mutually exclusive.

$G$ has an Euler path $\implies$ there is a vertex with odd degree $\implies$ $G$ cannot have an Euler circuit.

$G$ has an Euler circuit $\implies$ $G$ cannot have an Euler path.

If you mean what I think you mean, then you're correct.