l-facial edge colorings of graphs
Borut Lu\v{z}ar, Martina Mockov\v{c}iakov\'a, Roman Sot\'ak, Riste, \v{S}krekovski, Peter \v{S}ugerek
TL;DR
This paper proves that 7 colors are sufficient for 2-facial edge coloring of any plane graph, confirming a conjecture that 3l + 1 colors suffice for l-facial edge coloring.
Contribution
The authors prove the conjecture for l=2, establishing that 7 colors suffice for 2-facial edge coloring of plane graphs.
Findings
7 colors suffice for 2-facial edge coloring
Confirmed the conjecture for l=2
Established bounds for l-facial edge coloring
Abstract
An l-facial edge coloring of a plane graph is a coloring of the edges such that any two edges at distance at most l on a boundary walk of some face receive distinct colors. It is conjectured that 3l + 1 colors suffice for an l-facial edge coloring of any plane graph. We prove that 7 colors suffice for a 2-facial edge coloring of any plane graph and therefore confirm the conjecture for l = 2.
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