Was hoping if anyone had any way to prove the following claim using proof by contradiction
Let $G = (V, E)$ be a simple graph with at least one vertex, and let $G'$ be the graph formed by adding a new vertex $v$ and making it adjacent to every vertex in $V$.
Claim: $G$ has a Hamiltonian Path if and only if $G'$ has a Hamiltonian cycle.
I tried manipulating the definitions of each of the two (path vs. cycle), but didn't find much luck. Any thoughts?