AI 中文总结
本文证实Gastineau等的猜想，证明所有无爪次立方图为(1,1,2,3)-填充可着色，除单个图ℋ外的连通无爪次立方图为(1,1,3,3,3)-填充可着色，结果最优，证明依赖结构框架与类Hall匹配论证。
AI 中文摘要
对于非降正整数序列S=(a₁,a₂,…,aᵣ)，图G的S-填充着色是将V(G)划分为集合A₁,…,Aᵣ，使得对每个i∈{1,…,r}，Aᵢ中任意两个不同顶点的距离大于aᵢ。Gastineau和Togni在《Discrete Math.》339卷（2016年）第2461-2470页中提出疑问：除Petersen图外，所有次立方图是否都是(1,1,2,3)-填充可着色的。本文证明，所有无爪次立方图都是(1,1,2,3)-填充可着色的；此外，还证明所有连通无爪次立方图，除单个图ℋ外，都是(1,1,3,3,3)-填充可着色的，从而证实了前两位作者的一个猜想。这两个结果都是最优的。我们的证明基于无爪次立方图的骨架图与核心图构成的结构框架，结合类Hall匹配论证，将合适的3-填充构造简化为辅助二分图中的匹配问题。
英文摘要
For a non-decreasing sequence $S=(a_1,a_2,\ldots,a_r)$ of positive integers, an $S$-packing coloring of a graph $G$ is a partition of $V(G)$ into sets $A_1,\ldots,A_r$ such that any two distinct vertices in $A_i$ are at distance greater than $a_i$, for every $i\in\{1,\ldots,r\}$. Gastineau and Togni [\emph{Discrete Math.} 339 (2016), 2461--2470] asked whether every subcubic graph, except the Petersen graph, is $(1,1,2,3)$-packing colorable. In this paper, we prove that every claw-free subcubic graph is $(1,1,2,3)$-packing colorable. Moreover, we show that every connected claw-free subcubic graph, except a single graph $\mathcal{H}$, is $(1,1,3,3,3)$-packing colorable, thereby confirming a conjecture of the first two authors. Both results are best possible. Our proofs rely on a structural framework based on the skeleton and core graphs of a claw-free subcubic graph, together with a Hall-type matching argument that reduces the construction of suitable $3$-packings to a matching problem in an auxiliary bipartite graph.
