TITLE: How to find chromatic number of the $n$-dimensional hypercube $Q_n$?
QUESTION [5 upvotes]: How to find chromatic number the $n$-dimensional hypercube $Q_n$?
I know $\chi(Q_2)$=2 , $\chi(Q_3)$=2 , $\chi(Q_4)$=4

REPLY [22 votes]: Every hypercube is bipartite (and so the chromatic number is always 2). To see this, let $A$ be the set of all strings having an odd number of 1-bits and $B$ be the set of all strings having an even number of 1-bits. Since two strings are adjacent if and only if they differ in exactly one bit, it follows that there can be no edges between two vertices of $A$ or between two vertices of $B$.