Question: Are there any good video lectures available on Analytic Number Theory? I have a decent background of complex analysis but I have just started Analytic Number Theory.

Answer: 
Use the representation of the $k$-cube as a binary vector of length $k$.
For each set of binary digits $b2,\ldots,b\_k$ you have an edge between
$(0,b\_2,\ldots,b\_k)$ and $(1,b\_2,\ldots,b\_k)$.

Now show that this constitutes a perfect matching.
So this is how we tackle the problem: First, we put every vertex with an even degree sum in bipart $X$, and every vertex with an odd degree sum in bipart $Y$, which gives rise to a bipartition, as Leen proposed.

Then, given nod $V\_{x} = (0,b\_{2},b\_{3},\cdots ,b\_{k})$ form bipart $X$ and $V\_{y} = (1,b\_{2},b\_{3},\cdots ,b\_{k})$ from bipart $Y$ (where all $b\_{j}$'s are binary), we see that each pair of such vectors disagree on one single position in exactly $k$ ways; in other words each vertex $V\_{x}$ from $X$ is adjacent to exactly $k$ vertices $V\_{y}$ from $Y$ and vice versa, so that each vertex in the graph is of degree $k$, hence the graph is $k-regular$, as ml105 proposed.

Now what we have is a $bipartite$ $k-regular$ graph, for which the Marriage theorem guarantees the existence of a perfect matching.