On the $(1^2,2^4)$-packing edge-coloring of 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 Lužar (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.

Figures (13)

• Figure 1: A sharp example for the $(1^2, 2^4)$-packing edge-coloring of subcubic graphs.
• Figure 2: No Configuration $C_1$ (on the left) and its proof.
• Figure 3: No cycle (on the left) and its proof.
• Figure 4: No path of four edges and its proof.
• Figure 5: No two $K_{1,3}$s joined by an edge in $M_1 \cup M_2$ and its proof.

Theorems & Definitions (43)

• Conjecture 1: Hocquard, Lajou, and Lužar HLL2
• Conjecture 2: Gastineau and Togni GT1, Hocquard, Lajou, and Lužar HLL2
• Theorem 3
• Example 4
• Remark 5
• Definition 6
• Theorem 7: Kostochka and Yancey KY1
• Lemma 8
• proof
• Claim 8.1