Coloring Drawings of Graphs
Abstract
We consider cell colorings of drawings of graphs in the plane. Given a multi-graph G together with a drawing (G) in the plane with only finitely many crossings, we define a cell k-coloring of (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 K3,3-minor-free graphs.