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graphs that can be embedded into the plane, i.e. that can be drawn without crossings between the lines representing edges.
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Orientations of Planar Graphs
Such an orientation always exists, here is a proof.
Take your 2-edge-connected graph $G$, and consider its dual graph $D$. $D$ has a proper 4-coloring in which each face of $D$ contains at most 3 dif …