  We consider cell colorings of drawings of graphs in the plane. Given a
multi-graph $G$ together with a drawing $\Gamma(G)$ in the plane with only
finitely many crossings, we define a cell $k$-coloring of $\Gamma(G)$ to be a
coloring of the maximal connected regions of the drawing, the cells, with $k$
colors such that adjacent cells have different colors.
  By the $4$-color theorem, every drawing of a bridgeless graph has a cell
$4$-coloring. A drawing of a graph is cell $2$-colorable if and only if the
underlying graph is Eulerian. We show that every graph without degree 1
vertices admits a cell $3$-colorable drawing. This leads to the natural
question which abstract graphs have the property that each of their drawings
has a cell $3$-coloring. We say that such a graph is universally cell
$3$-colorable. We show that every $4$-edge-connected graph and every graph
admitting a nowhere-zero $3$-flow is universally cell $3$-colorable. We also
discuss circumstances under which universal cell $3$-colorability guarantees
the existence of a nowhere-zero $3$-flow. On the negative side, we present an
infinite family of universally cell $3$-colorable graphs without a nowhere-zero
$3$-flow. On the positive side, we formulate a conjecture which has a
surprising relation to a famous open problem by Tutte known as the
$3$-flow-conjecture. We prove our conjecture for subcubic and for
$K_{3,3}$-minor-free graphs.
