On S-packing edge-colorings of cubic graphs

TL;DR

This paper investigates S-packing edge-colorings of cubic graphs, establishing new coloring bounds for graphs with specific properties and proposing open problems for future research.

Contribution

It introduces new results on S-packing edge-colorings of cubic graphs, including bounds for graphs with 2-factors and those with bounded oddness.

Findings

Cubic graphs with a 2-factor are (1,1,1,3,3)-packing edge-colorable

Cubic graphs with a 2-factor are (1,1,1,4,4,4,4,4)-packing edge-colorable

Cubic graphs with a 2-factor are (1,1,2,2,2,2,2)-packing edge-colorable

Abstract

Given a non-decreasing sequence S = (s 1,s 2,. .. ,s k) of positive integers, an S-packing edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 ,X 2,. .. ,X k } such that for each 1 $\le$ i $\le$ k, the distance between two distinct edges e, e ' $\in$ X i is at least s i + 1. This paper studies S-packing edge-colorings of cubic graphs. Among other results, we prove that cubic graphs having a 2-factor are (1,1,1,3,3)-packing edge-colorable, (1,1,1,4,4,4,4,4)-packing edge-colorable and (1,1,2,2,2,2,2)-packing edge-colorable. We determine sharper results for cubic graphs of bounded oddness and 3-edge-colorable cubic graphs and we propose many open problems.