Coloring Drawings of Graphs
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We consider face-colorings of drawings of graphs in the plane. Given a multi-graph G together with a drawing Γ(G) in the plane with only finitely many crossings, we define a face-k-coloring of Γ(G) to be a coloring of the maximal connected regions of the drawing, the faces, with k colors such that adjacent faces have different colors. By the 4-color theorem, every drawing of a bridgeless graph has a face-4-coloring. A drawing of a graph is facially 2-colorable if and only if the underlying graph is Eulerian. We show that every graph without degree 1 vertices admits a 3-colorable drawing. This leads to the natural question which graphs G have the property that each of its drawings has a 3-coloring. We say that such a graph G is facially 3-colorable. We derive several sufficient and necessary conditions for this property: we show that every 4-edge-connected graph and every graph admitting a nowhere-zero 3-flow is facially 3-colorable. We also discuss circumstances under which facial 3-colorability guarantees the existence of a nowhere-zero 3-flow. On the negative side, we present an infinite family of facially 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.
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