稀疏亚立方图的(1,2⁴)和(1,2⁵)-填充边着色
On $(1,2^4)$ and $(1,2^5)$-packing edge-coloring of sparse subcubic graphs
- Konkuk University（建国大学）
- Korea Institute for Advanced Study (KIAS)（韩国高等研究院）
- Xi’an Jiaotong-Liverpool University（西交利物浦大学）
- Guizhou Minzu University（贵州民族大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对亚立方平面图，确定了使这类图存在(1,2⁵)-、(1,2⁴)-填充边着色的最小girth阈值的范围，证明两阈值均为有限值。
AI中文摘要：
图G的(1^j,2^k)-填充边着色是将边集E(G)划分为j个匹配和k个诱导匹配。Hocquard、Lajou和Lužar发现了 girth 为3的亚立方平面图，其不存在(1,2⁵)-填充边着色，并猜想所有亚立方平面图都存在(1,2⁶)-填充边着色。我们还注意到，对每个固定正整数k，存在girth为k的亚立方平面图，其不可进行(1,2³)-填充边着色。因此自然要探究最小正整数k₁，使得所有girth至少为k₁的亚立方平面图都存在(1,2⁵)-填充边着色；以及最小正整数k₂，使得所有girth至少为k₂的亚立方平面图都存在(1,2⁴)-填充边着色。本文证明k₁和k₂均为有限值，实际有5≤k₁≤12，6≤k₂≤16。
英文摘要：
A $(1^j,2^k)$-packing edge-coloring of a graph $G$ is a partition of the edge set $E(G)$ into $j$ matchings and $k$ induced matchings. Hocquard, Lajou, and Lu\v zar found a subcubic planar graph of girth $3$ that has no $(1,2^5)$-packing edge-coloring and also conjectured that every subcubic planar graph has a $(1,2^6)$-packing edge-coloring. We also notice that for every fixed positive integer $k$ there exists a subcubic planar graph with girth $k$ that is not $(1,2^3)$-packing edge-colorable. It is natural to consider what is the minimum positive integer $k_1$ such that every subcubic planar graph with girth at least $k_1$ is $(1,2^5)$-packing edge-colorable. Furthermore, we also consider what is the minimum positive integer $k_2$ such that every subcubic planar graph with girth at least $k_2$ is $(1,2^4)$-packing edge-colorable. In this paper, we show both $k_1$ and $k_2$ are finite, and in fact $5 \le k_1 \le 12$ and $6 \le k_2 \le 16$.
