  Bicliques are complements of bipartite graphs; as such each consists of two
cliques joined by a number of edges. In this paper we study algebraic aspects
of the chromatic polynomials of these graphs. We derive a formula for the
chromatic polynomial of an arbitrary biclique, and use this to give certain
conditions under which two of the graphs have chromatic polynomials with the
same splitting field. Finally, we use a subfamily of bicliques to prove the
cubic case of the $\alpha +n$ conjecture, by showing that for any cubic integer
$\alpha$, there is a natural number $n$ such that $\alpha +n$ is a chromatic
root.
