Union of Hamiltonian graphs

Question

If we have two Hamiltonian graphs =(,) and ′=(,′) that are on the same set of ≥5 vertices and do not share any edges. Is the union of ′and also Hamiltonian?

Answer 1

Let's call $G$ the graph which is the union of $H$ and $H'$.

The vertices of $G$ are V and the edges of $G$ are $E \cup E'$.

$H$ is Hamiltonian so there is a cycle $C=\{e_1, e_2, ..., e_n\}$ which (a) visits every vertex in $V$; and (b) each $e_i$ is in $E$.

So let's look at $C$ in $G$. It is still true that it (a) visits every vertex in $V$; and (b) each $e_i$ is in $E \cup E'$. So $C$ is a cycle in $G$ so $G$ is Hamiltonian.