Coloring Drawings of Graphs

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.