Let $G=(V,E)$ be a simple, undirected graph. A *matching* is a subset $M\subseteq E$ such that all members of $M$ are pairwise disjoint; moreover we call $M$ *perfect* if $\bigcup M = V$.

The 1-factorization conjecture states that if the minimum degree $\delta(G)$ is at least $n$ and $|V(G)| = 2n$, then $G$ has a perfect matching.

Dirac's theorem states that if $\delta(G)\geq n$ and $|V(G)| = 2n$, then $G$ has Hamiltonian cycle.

Given $G$ with $\delta(G)\geq n$ and $|V(G)| = 2n$, we can't we just take a Hamiltonian cycle and pick every second edge to get the desired perfect matching?