A graph $G$ is $k$-edge colorable if there exists a function $f\colon E(G) → [k]$ s.t $f(e) \neq f(e')$ whenever $e$ and $e'$ share a vertex. A Hamiltonian cycle in an $n$-vertex graph is a sub-graph isomorphic to $C_n$. Finally, a graph $G$ is $d$-regular if $\deg_G(v) = d$ for every $v ∈ V (G)$.
Prove that every 3-regular graph that contains a Hamiltonian cycle is always 3-edge colorable. Also, does this apply more generally to say that any $n$-regular graph with a Hamiltonian cycle is $n$-edge colorable?