3-facial edge-coloring of plane graphs

TL;DR

This paper proves that for plane graphs, an $ ext{3-facial}$ edge-coloring can be achieved with at most 10 colors, confirming a conjecture for the case when $ ext{ell} = 3$.

Contribution

The paper establishes the conjecture that at most $3 ext{ell} + 1$ colors suffice for $ ext{ell}$-facial edge-coloring of plane graphs, specifically proving it for $ ext{ell} = 3$.

Findings

Confirmed the conjecture for $ ext{ell} = 3$

Established an upper bound of 10 colors for 3-facial edge-coloring

Extended the understanding of facial edge-colorings in plane graphs

Abstract

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