Final answer:
An Euler path is a path that visits every edge of a graph exactly once, while an Euler cycle is an Euler path that starts and ends at the same vertex. The main difference between the two is whether it is a closed loop or not.
Step-by-step explanation:
An Euler path is a path that visits every edge of a graph exactly once, while an Euler cycle is an Euler path that starts and ends at the same vertex. In other words, an Euler path is a trail, while an Euler cycle is a closed loop.
For example, consider a graph with vertices A, B, C, D, and E, and edges AB, BC, CD, DE, and EA. If a path exists that visits all the edges exactly once, such as A -> B -> C -> D -> E -> A, it is an Euler cycle. If the path does not end at the same vertex, it is an Euler path but not an Euler cycle.
In summary, the main difference between an Euler path and an Euler cycle is whether it starts and ends at the same vertex.