TITLE: A graph of order $2n$ for which all vertices have degree $\geq n$ may be partitioned into adjacent pairs.
QUESTION [6 upvotes]: Suppose that $G$ is a graph with $2n$ vertices for which every vertex has degree at least $n$. Prove that we can partition $V(G)$ into pairs such that the two vertices in each pair are adjacent.

REPLY [2 votes]: Dirac's theorem gives us a Hamiltonian cycle $v_1v_2v_3\ldots v_{2n-1}v_{2n}v_1$, so $$\left\{\left(v_i,v_{i+1}\right):i=1,\ldots,2n-1\right\}$$
is one such set of pairs.