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$ From the definition of isomorphism, prove that `G cong H iff bar(G) cong bar(H)`.
Proof: Suppose `G cong H`

If `G cong H`, there exists a bijection `f: V(G) to V(H)` such that `uv` in `E(G)` if and only if `f(u)f(v) in E(H)`.

Suppose `bar(G) cong bar(H)`

q.e.d
$ Prove or disprove: If every vertex of a simple graph `G` has 2 degree, then `G` is a cycle.

...[under development:)]

$ Prove that a graph with more than six vertices of odd degree cannot be decomposed into three paths.

...[under development:)]