Coloring Drawings of Graphs

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