is it necessary to cover all the verticies in eular path? I was going through graph theory and came across the term Euler path or some people prefer Euler trail as vertices can repeat.

According to the definition from wiki (https://en.wikipedia.org/wiki/Eulerian_path), Euler path is defined as under


  In graph theory, a Eulerian trail (or Eulerian path) is a trail in a graph which visits every edge exactly once.


Following are the conditions for Euler path,


  An undirected graph (G) has a Eulerian path if and only if every vertex has even degree except 2 vertices which will have odd degree, and all of its vertices with nonzero degree belong to a single connected component.


So as per the definition, I need to cover all edges in a connected graph with non zero degrees. 

The same information is stated here at geeks for geeks(https://www.geeksforgeeks.org/eulerian-path-and-circuit/).

so if I need to only consider vertices with non zero degrees then graph (G)can be disconnected?

Note: These are the other information I collected from the wiki page

1)Finite connected graph (with vertices of even degree except 2 or 0 with the odd degree) will have a Euler path.
2)But Euler path can also be present in the disconnected graph as shown in the following picture



3) Doubt does following graph have Euler path,

My answer ,No as all vertices are not in same connected component.
It really comes down to your definition of an Euler trail. The one I'm familiar with is similar to the one on Wikipedia:


  An Euler trail is a trail (path that allows repeats) that uses every edge exactly once.


With this definition, an Euler trail doesn't have to touch every vertex: if there's a vertex with no edges, the Euler trail doesn't have to go anywhere near it.

However, if there's a vertex with positive degree, and another vertex with positive degree, and they're not (weakly) connected, then there can be no Euler trail. Because if you use the edges on one, there's no way to reach the edges on the other.