TITLE: prove $n$-cube is bipartite
QUESTION [9 upvotes]: prove $n$-cube is a bipartite graph for all $n\ge1$ 
This is a problem in my textbook and I cannot figure it out at all and have a test on graph theory tomorrow any help would be appreciated since I am not very good with proofs

REPLY [10 votes]: When labeling your $n$-cube, you can assign the vertices strings of length $n$ from $(00..0)$ to $(11..1)$. For example a $2$-cube (or square) would be:
$$00\ 01\\10\ 11$$
Now you can define two subesets of your graph $G= (X,Y)$. 
Where $X=${vertices who have an even number of $1$'s} and $Y=${vetices who have an odd number of $1$'s}.
By creating an edges {$1,0$} or {$0,1$} starting in $X$ and ending in $Y$ all edges are accounted for and therefore, the graph is bipartite.