Coloring drawings of graphs (Q2073312)

Summary: 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.