Strong edge coloring of planar graphs

TL;DR

This paper improves bounds on the number of colors needed for strong edge coloring of planar graphs, showing that higher girth allows for fewer colors, with specific results for cubic graphs.

Contribution

It establishes new upper bounds for strong edge coloring of planar graphs based on girth and degree, refining previous known bounds.

Findings

3D + 6 colors suffice for girth 6 graphs

3D colors suffice for girth ≥ 7

Cubic planar graphs with girth ≥ 6 can be colored with at most 9 colors

Abstract

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree D has a strong edge coloring with at most 4D + 4 colors. We show that 3D + 6 colors suffice if the graph has girth 6, and 3D colors suffice if the girth is at least 7. Moreover, we show that cubic planar graphs with girth at least 6 can be strongly edge colored with at most 9 colors.