For non-decreasing sequence of integers $S=(a_1,a_2, \dots, a_k)$, an $S$-packing coloring of $G$ is a partition of $V(G)$ into $k$ subsets $V_1,V_2,\dots,V_k$ such that the distance between any two distinct vertices $x,y \in V_i$ is at least $a_{i}+1$, $1\leq i\leq k$. We consider the $S$-packing coloring problem on subclasses of subcubic graphs: For $0\le i\le 3$, a subcubic graph $G$ is said to be $i$-saturated if every vertex of degree 3 is adjacent to at most $i$ vertices of degree 3. Furthermore, a vertex of degree 3 in a subcubic graph is called heavy if all its three neighbors are of degree 3, and $G$ is said to be $(3,i)$-saturated if every heavy vertex is adjacent to at most $i$ heavy vertices. We prove that every 1-saturated subcubic graph is $(1,1,3,3)$-packing colorable and $(1,2,2,2,2)$-packing colorable. We also prove that every $(3,0)$-saturated subcubic graph is $(1,2,2,2,2,2)$-packing colorable.

翻译：对于非递减整数序列$S=(a_1,a_2, \dots, a_k)$，图$G$的$S$-包装着色是将$V(G)$划分为$k$个子集$V_1,V_2,\dots,V_k$，使得对于任意两个不同顶点$x,y \in V_i$，其距离至少为$a_{i}+1$，其中$1\leq i\leq k$。本文研究子三次图子类上的$S$-包装着色问题：对于$0\le i\le 3$，若子三次图$G$中每个3度顶点至多与$i$个3度顶点相邻，则称$G$为$i$-饱和的。此外，在子三次图中，若一个3度顶点的三个邻点均为3度顶点，则称该顶点为重顶点；若每个重顶点至多与$i$个重顶点相邻，则称$G$为$(3,i)$-饱和的。我们证明：所有1-饱和子三次图均具有$(1,1,3,3)$-包装着色与$(1,2,2,2,2)$-包装着色；同时证明所有$(3,0)$-饱和子三次图均具有$(1,2,2,2,2,2)$-包装着色。