Let's consider undirected simple and connected graph.
I want to partition the vertex set into a minimal number of disjoint subsets such that the induced graph of each subset has an even number of edges?
I observed that if we have at least one odd degree vertex then we can move it to a different set then set of all other vertices. In this case we will have at most 2 disjoint sets.
Now when we don't have any odd degree vertex we must have eulerian cycle.
Now we can find smallest odd length cycle. All vertices not in cycle will now receive same set(let say set number $1$). And we will pick two adjacent vertices in our cycle then all other vertex in cycle can now receive same set(let say $2$) because this two vertex will eliminate 3 edges from cycle. Now we have to check whether our selected 2 vertices have any common edges to vertex not in cycle if not we can assign them same set as non cycle vertex(i.e set $1$). (here noncyclic vertex is vertex not included in our smallest odd length cycle.)
I want to know whether my approach is correct or not. Any help and suggestion would be very appreciated.