Every subcubic multigraph is $(1,2^7)$-packing edge-colorable

TL;DR

This paper proves that every subcubic multigraph can be edge-colored with a specific packing scheme, confirming a recent conjecture and extending the result from simple graphs to multigraphs.

Contribution

It confirms the conjecture that all subcubic graphs are $(1,2^7)$-packing edge-colorable and extends this result to multigraphs.

Findings

Every subcubic multigraph is $(1,2^7)$-packing edge-colorable.

Confirmed the conjecture for simple graphs and extended it to multigraphs.

Strengthens understanding of packing edge-colorings in low-degree graphs.

Abstract

For a non-decreasing sequence $S = (s_1, \ldots, s_k)$ of positive integers, an $S$-packing edge-coloring of a graph $G$ is a decomposition of edges of $G$ into disjoint sets $E_1, \ldots, E_k$ such that for each $1 \le i \le k$ the distance between any two distinct edges $e_1, e_2 \in E_i$ is at least $s_i+1$. The notion of $S$-packing edge-coloring was first generalized by Gastineau and Togni from its vertex counterpart. They showed that there are subcubic graphs that are not $(1,2,2,2,2,2,2)$-packing (abbreviated to $(1,2^6)$-packing) edge-colorable and asked the question whether every subcubic graph is $(1,2^7)$-packing edge-colorable. Very recently, Hocquard, Lajou, and Lu\v{z}ar showed that every subcubic graph is $(1,2^8)$-packing edge-colorable and every $3$-edge colorable subcubic graph is $(1,2^7)$-packing edge-colorable. Furthermore, they also conjectured that every subcubic graph is $(1,2^7)$-packing edge-colorable. In this paper, we confirm the conjecture of Hocquard, Lajou, and Lu\v{z}ar, and extend it to multigraphs.