Coloring Drawings of Graphs

Abstract

We consider cell colorings of drawings of graphs in the plane. Given a multi-graphGtogether with a drawing Γ(G) in the plane with only finitely many crossings,we define acellk-coloringof Γ(G) to be a coloring of the maximal connected regionsof the drawing, thecells, withkcolors 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 isEulerian. We show that every graph without degree 1 vertices admits a cell 3-colorable drawing. This leads to the natural question whichabstractgraphs havethe property that each of their drawings has a cell 3-coloring. We say that such agraph isuniversally cell3-colorable. We show that every 4-edge-connected graphand every graph admitting anowhere-zero3-flowis universally cell 3-colorable. Wealso discuss circumstances under which universal cell 3-colorability guarantees theexistence of a nowhere-zero 3-flow. On the negative side, we present an infinitefamily of universally cell 3-colorable graphs without a nowhere-zero 3-flow. On thepositive side, we formulate a conjecture which has a surprising relation to a famousopen problem by Tutte known as the 3-flow-conjecture. We prove our conjecturefor subcubic and forK3,3-minor-free graphs.