Two graphs G and F are isomorphic iff there is a bijection between E(G) an E(F)

Question

I want to check if the following statement is true:

G and F are isomorphic graphs iff there exists a bijection f: E(G) -> E(F)

Answer 1

No, it is not enough. It must preserve adjacency.

Say $G= \{1,2,3\}$ with edges $12$ and $13$

and $ F= \{a,b,c,d\}$ with edges $ab$ and $cd$.

Clearly we have a bijection between edges but the graphs are not isomorphic.