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We obtain a complete classification of all simple closed geodesics on the eight convex deltahedra by solving a related graph coloring problem. Geodesic segments in the neighborhood of each deltahedron vertex produce a limited number of crossing angles with deltahedron edges. We define a coloring on the edge graph of a deltahedron based on these angles, and we show that the set of graph colorings compatible with edge-colorings of the neighborhood graphs of radius one classifies all possible simple closed geodesics on all convex deltahedra.