Packing edge-colorings of subcubic outerplanar graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915072307101696 |
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| author | Li, Sijin Li, Yifan Liu, Xujun |
| author_facet | Li, Sijin Li, Yifan Liu, Xujun |
| contents | For a sequence $S = (s_1, s_2, \ldots, s_k)$ of non-decreasing positive integers, an $S$-packing edge-coloring (S-coloring) of a graph $G$ is a partition of $E(G)$ into $E_1, E_2, \ldots, E_k$ such that the distance between each pair of distinct edges $e_1,e_2 \in E_i$, $1 \le i \le k$, is at least $s_i + 1$. In particular, a $(1^{\ell},2^k)$-coloring is a partition of $E(G)$ into $\ell$ 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 $(1,2^6)$-coloring and a $(1^2,2^3)$-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 $(1,2^5)$-coloring and a $(1^2,2^3)$-coloring. Our results are best possible since we found subcubic outerplanar graphs with no $(1,2^4)$-coloring and no $(1^2,2^2)$-coloring respectively. Furthermore, we explore the question "What is the largest positive integer $k_1$ and $k_2$ such that every subcubic outerplanar graph is $(1,2^4,k_1)$-colorable and $(1^2,2^2,k_2)$-colorable?". We prove $3 \le k_1 \le 6$ and $3 \le k_2 \le 4$. We also consider the question "What is the largest positive integer $k_1'$ and $k_2'$ such that every $2$-connected subcubic outerplanar graph is $(1,2^3,k_1')$-colorable and $(1^2,2^2,k_2')$-colorable?". We prove $k_1' = 2$ and $3 \le k_2' \le 11$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05720 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Packing edge-colorings of subcubic outerplanar graphs Li, Sijin Li, Yifan Liu, Xujun Combinatorics For a sequence $S = (s_1, s_2, \ldots, s_k)$ of non-decreasing positive integers, an $S$-packing edge-coloring (S-coloring) of a graph $G$ is a partition of $E(G)$ into $E_1, E_2, \ldots, E_k$ such that the distance between each pair of distinct edges $e_1,e_2 \in E_i$, $1 \le i \le k$, is at least $s_i + 1$. In particular, a $(1^{\ell},2^k)$-coloring is a partition of $E(G)$ into $\ell$ 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 $(1,2^6)$-coloring and a $(1^2,2^3)$-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 $(1,2^5)$-coloring and a $(1^2,2^3)$-coloring. Our results are best possible since we found subcubic outerplanar graphs with no $(1,2^4)$-coloring and no $(1^2,2^2)$-coloring respectively. Furthermore, we explore the question "What is the largest positive integer $k_1$ and $k_2$ such that every subcubic outerplanar graph is $(1,2^4,k_1)$-colorable and $(1^2,2^2,k_2)$-colorable?". We prove $3 \le k_1 \le 6$ and $3 \le k_2 \le 4$. We also consider the question "What is the largest positive integer $k_1'$ and $k_2'$ such that every $2$-connected subcubic outerplanar graph is $(1,2^3,k_1')$-colorable and $(1^2,2^2,k_2')$-colorable?". We prove $k_1' = 2$ and $3 \le k_2' \le 11$. |
| title | Packing edge-colorings of subcubic outerplanar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.05720 |