On the $(1^2,2^4)$-packing edge-coloring of subcubic graphs

TL;DR

This paper proves that all large connected subcubic graphs with over 70 vertices can be edge-colored with a specific packing scheme, confirming a longstanding conjecture for these graphs.

Contribution

It confirms the conjecture that every connected subcubic graph with more than 70 vertices is $(1^2,2^4)$-packing edge-colorable, a result previously unproven.

Findings

Confirmed the conjecture for graphs with more than 70 vertices

Identified the sharpness of the result with counterexamples for $(1^2,2^3)$-colorability

Extended understanding of edge-coloring in subcubic graphs

Abstract

An induced matching in a graph $G$ is a matching such that its end vertices also induce a matching. A $(1^{\ell}, 2^k)$-packing edge-coloring of a graph $G$ is a partition of its edge set into disjoint unions of $\ell$ matchings and $k$ induced matchings. Gastineau and Togni (2019), as well as Hocquard, Lajou, and Lu\v{z}ar (2022), have conjectured that every subcubic graph is $(1^2,2^4)$-packing edge-colorable. In this paper, we confirm that their conjecture is true (for connected subcubic graphs with more than $70$ vertices). Our result is sharp due to the existence of subcubic graphs that are not $(1^2,2^3)$-packing edge-colorable.