  Albertson conjectured that if graph $G$ has chromatic number $r$, then the
crossing number of $G$ is at least that of the complete graph $K_r$. This
conjecture in the case $r=5$ is equivalent to the four color theorem. It was
verified for $r=6$ by Oporowski and Zhao.
  In this paper, we prove the conjecture for $7 \leq r \leq 12$ using results
of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the
number of edges in critical graphs, together with lower bounds by Pach et.al.
on the crossing number of graphs in terms of the number of edges and vertices.
