Answer : Yes
Disclaimer : Before you read my explanations , I found some resources out on internet which have different definitions for what a Eulerian Circuit and Eulerian Path is… like this and this… I won’t argue the reliability of those sources. The following answer is coherent with Definitions present on wikipedia.
Eulerian path is a trail in graph that visits every edge exactly once.
Eulerian Circuit is an Eulerian Path which starts and ends on the same vertex.
$\therefore$ Every Eulerian Circuit is also an Eulerian path. So any graph that has Eulerian Circuit has Eulerian Paths.
The converse may not be True , means Even if a graph has an Eulerian Path , Eulerian Circuit is not guaranteed to exist.
Furthermore following points are true irrespective of source of information :
- Connected graph with every vertex with even degree is both necessary and sufficient condition for Eulerian Circuit.
- Connected graph with exactly 2 vertices with odd degree have Eulerian Path but not an Eulerian Circuit.
- Anything Else neither have Eulerian Path nor Eulerian Circuit.
Example :
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In above graph the trail $V_1E_1V_2E_2V_3E_3V_4E_4V_5E_5V_1$ is both an Eulerian Path as well as Eulerian Circuit. As it visits all edges and starts and ends on same vertex .
Now if any one of the $5$ edge was absent like in example below :
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In above graph vertex $V_1$ and $V_5$ are only two vertices with odd degree so the trail $V_1E_1V_2E_2V_3E_3V_4E_4V_5$ is an Eulerian Path and there is no Eulerian Circuit.
In such graphs the Eulerian Path starts on one of the odd degree vertex and ends on the other odd degree vertex.