Given a graph G and a nondecreasing sequence \(S=(s_1,\ldots ,s_k)\) of positive integers, the mapping \(c:V(G)\longrightarrow \{1,\ldots ,k\}\) is called an S -packing coloring of G if for any two distinct vertices x and y in \(c^{-1}(i)\), the distance between x and y is greater than \(s_i\). The smallest integer k such that there exists a \((1,2,\ldots ,k)\)-packing coloring of a graph G is called the packing chromatic number of G , denoted \(\chi _{\rho }(G)\). The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by 7. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a (1, 2, 2, 2)-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a (1, 2, 2, 2)-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a (1, 2, 2, 3)-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an S -packing coloring for \(S=(1,3,\ldots ,3)\), where 3 appears \(\Delta \) times (\(\Delta \) being the maximum degree of vertices), and this property does not hold if one of the integers 3 is replaced by 4 in the sequence S .

中文翻译：

---

次三次外平面图的填充色

给定图 G 和正整数的非递减序列\（S =（s_1，\ ldots，s_k）\），映射\（c：V（G）\ longrightarrow \\ {1，\ ldots，k \} \）如果对于\（c ^ {-1}（i）\）中任意两个不同的顶点 x 和 y ， x 和 y 之间的距离 大于\（s_i \），则称为 G 的 S 填充着色 。的最小整数 ķ 使得存在一个\（（1,2，\ ldots，K）\） -的曲线图的包装着色 ģ 被称为的填料色数 ģ ，表示为\（\ chi _ {\ rho}（G）\）。在较早的几篇论文中研究了亚三次（平面）图类中填充色数的有界性问题；最近已经确定，在所有次三次图的类中，不变性是无界的。在本文中，我们证明了任何2个连通的二部分次立方外平面图的填充色数均以7为界。此外，我们证明了每个次立方无三角形外平面图都具有（1、2、2、2）堆积着色，并且存在一个带有不容许（1、2、2、2、2）填充着色​​的三角形的次三次外平面图。此外，存在一个不包含（1,2,2,2,3）堆积着色的无亚三次三角形的外平面图。对于二分外平面图，显示了类似的二分法：每个这样的图都承认一个 \（S =（1,3，\ ldots，3）\）的 S 包装着色，其中3出现\（\ Delta \）倍（\（\ Delta \）是最大顶点度），并且此属性如果在序列 S 中将整数3之一替换为4，则不成立 。