ℓℓ-facial edge colorings of graphs

An ℓℓ-facial edge coloring   of a plane graph is a coloring of the edges such that any two edges at distance at most ℓℓ on a boundary walk of any face receive distinct colors. It is conjectured that 3ℓ+1 colors suffice for an ℓℓ-facial edge coloring of any plane graph. We prove that 77 colors suffice for a 22-facial edge coloring of any plane graph and therefore we confirm the conjecture for ℓ=2ℓ=2.