I have the following the statement and I have to prove whether it is true or not.
Given a planar and Hamiltonian graph $\mathbb{G} = (V, E)$, show that we can partition $E = E_1 \cup E_2 = E$ so that $G_1 = (V, E_1)$ and $G_2 = (V, E_2)$ are both connected and outerplanar graphs.
I have tested the statement for few graphs and it seems correct to me, but I am not sure how can I use the property of the Hamiltonian cycle to decompose the graph into two sub-graphs.
I appreciate any hint!
