On $(1^2,2^2)$-packing edge-coloring of sparse subcubic graphs
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915864476909568 |
|---|---|
| author | Liu, Xujun Yang, Jiacheng Zhang, Xin |
| author_facet | Liu, Xujun Yang, Jiacheng Zhang, Xin |
| contents | For positive integers $\ell$ and $k$, a $(1^\ell, 2^k)$-packing edge-coloring of a graph $G$ is a partition of $E(G)$ into $\ell$ matchings and $k$ induced matchings. A graph is $d$-irregular if it has no adjacent vertices of degree $d$. Yang and Wu proved that every $3$-irregular subcubic graph admits a $(1,2^4)$-packing edge-coloring, which answered an open question of Hocquad, Lajou, and Lu\v zar in the affirmative. In this paper, we prove an analogue result that every $3$-irregular subcubic multigraph is $(1^2,2^2)$-packing edge-colorable. Our result is sharp since there are $3$-irregular subcubic graphs that are not $(1,2^3)$-packing edge-colorable and $(1^2,2)$-packing edge-colorable, respectively.
Hocquad, Lajou, and Lu\v zar conjectured that every subcubic planar graph is $(1^2,2^3)$-packing edge-colorable. Furthermore, they found a subcubic planar graph with girth $3$ that is not $(1^2,2^2)$-packing edge-colorable. For every fixed integer $k \ge 3$, we found graphs with girth $k$ that are not $(1^2,2)$- and not $(1,2^3)$-packing edge-colorable. It is natural to consider the question "what is the minimum positive integer $g$ such that every subcubic planar graph with girth at least $g$ is $(1^2,2^2)$-packing edge-colorable?". We prove $g$ is finite and in fact $g \le 20$. We also provide an example showing $g \ge 6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14398 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $(1^2,2^2)$-packing edge-coloring of sparse subcubic graphs Liu, Xujun Yang, Jiacheng Zhang, Xin Combinatorics For positive integers $\ell$ and $k$, a $(1^\ell, 2^k)$-packing edge-coloring of a graph $G$ is a partition of $E(G)$ into $\ell$ matchings and $k$ induced matchings. A graph is $d$-irregular if it has no adjacent vertices of degree $d$. Yang and Wu proved that every $3$-irregular subcubic graph admits a $(1,2^4)$-packing edge-coloring, which answered an open question of Hocquad, Lajou, and Lu\v zar in the affirmative. In this paper, we prove an analogue result that every $3$-irregular subcubic multigraph is $(1^2,2^2)$-packing edge-colorable. Our result is sharp since there are $3$-irregular subcubic graphs that are not $(1,2^3)$-packing edge-colorable and $(1^2,2)$-packing edge-colorable, respectively. Hocquad, Lajou, and Lu\v zar conjectured that every subcubic planar graph is $(1^2,2^3)$-packing edge-colorable. Furthermore, they found a subcubic planar graph with girth $3$ that is not $(1^2,2^2)$-packing edge-colorable. For every fixed integer $k \ge 3$, we found graphs with girth $k$ that are not $(1^2,2)$- and not $(1,2^3)$-packing edge-colorable. It is natural to consider the question "what is the minimum positive integer $g$ such that every subcubic planar graph with girth at least $g$ is $(1^2,2^2)$-packing edge-colorable?". We prove $g$ is finite and in fact $g \le 20$. We also provide an example showing $g \ge 6$. |
| title | On $(1^2,2^2)$-packing edge-coloring of sparse subcubic graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.14398 |