Is it possible that exactly one of $G$ and $H$ has an Euler trail, if both graphs are homeomorphic?

Question

Let $G$ and $H$ be loop-free homeomorphic undirected graphs with no isolated vertices. Is it possible that exactly one of $G$ and $H$ has an Euler trail?

My approach is that if graph $G$ has an Euler trail then all of its homeomorphic graphs have Euler trail because each subdivison of $G$ creates a vertex with even degree; so it does not break property of Euler trailer as there will always be two vertices with odd degree.

Answer 1

To mark this question answered: you are right.