3-facial edge-coloring of plane graphs

Abstract

An ℓ-facial edge-coloring of a plane graph is a coloring of its edges such that any two edges at distance at most ℓ on a boundary walk of any face receive distinct colors. It is the edge-coloring variant of the ℓ-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It is conjectured that at most 3ℓ+1 colors suffice for an ℓ-facial edge-coloring of any plane graph. The conjecture has only been confirmed for ℓ≤2, and in this paper, we prove its validity for ℓ=3.