Packing edge-colorings of subcubic outerplanar graphs

For a sequence S=(s₁,s₂,…,s_k) of non-decreasing positive integers, an S-packing edge-coloring (S-coloring) of a graph G is a partition of E(G) into E₁,E₂,…,E_k such that the distance between each pair of distinct edges e₁,e₂∈E_i, 1≤i≤k, is at least s_i+1. In particular, a (1^ℓ,2^k)-coloring is a partition of E(G) into ℓ matchings and k induced matchings, and it can be viewed as intermediate colorings between proper and strong edge-colorings. Hocquard, Lajou, and Lužar conjectured that every subcubic planar graph has a (1,2⁶)-coloring and a (1²,2³)-coloring. In this paper, we confirm the conjecture of Hocquard, Lajou, and Lužar for subcubic outerplanar graphs by showing every subcubic outerplanar graph has a (1,2⁵)-coloring and a (1²,2³)-coloring. Our results are the best possible since we found subcubic outerplanar graphs with no (1,2⁴)-coloring and no (1²,2²)-coloring respectively. Furthermore, we explore the question “What are the largest positive integer k₁ and k₂ such that every subcubic outerplanar graph is (1,2⁴,k₁)-colorable and (1²,2²,k₂)-colorable?”. We prove 3≤k₁≤6 and 3≤k₂≤4. We also consider the question “What are the largest positive integer k₁^′ and k₂^′ such that every 2-connected subcubic outerplanar graph is (1,2³,k₁^′)-colorable and (1²,2²,k₂^′)-colorable?”. We prove k₁^′=2 and 3≤k₂^′≤11.