  Woodall (and Seymour independently) in 2001 proposed a conjecture that every
graph $G$ contains every complete bipartite graph on $\chi(G)$ vertices as a
minor, where $\chi(G)$ is the chromatic number of $G$. In this paper, we prove
that for each positive integer $\ell$ with $2\ell \leq \chi(G)$, each graph $G$
with independence number two contains a $K^{\ell}_{\ell,\chi(G)-\ell}$-minor,
implying that Seymour and Woodall's conjecture holds for graphs with
independence number two, where $K^{\ell}_{\ell,\chi(G)-\ell}$ is the graph
obtained from $K_{\ell,\chi(G)-\ell}$ by making every pair of vertices on the
side of the bipartition of size $\ell$ adjacent.
