仙人掌图的两距离染色与列表两距离染色：次立方情形与C5障碍
Two-distance and list-two-distance coloring of cacti: the subcubic case and the C5 obstruction
浏览论文内容
中文总结 AI 辅助
本文精确确定了仙人掌图的两距离色数与列表两距离色数，证明二者相等，并给出次立方情形的完整分类，以5-圈为唯一障碍，同时验证了列表平方染色猜想的一个正例。
中文摘要 AI 辅助
图$G$的平方$G^2$连接距离至多为2的两个顶点；$G^2$的正常染色称为$G$的2-距离染色。对于仙人掌图$G$（每条边至多位于一个环上），我们精确确定了2-距离色数$\chi(G^2)$和选择数$\mathrm{ch}(G^2)$：它们总是相等，且公共值为：若$\Delta\ge4$则为$\Delta+1$；若$\Delta=3$且$G$没有等于$C_5$的块则为4；若$\Delta=3$且$G$含有$C_5$块则为5；若$\Delta\le2$则为经典值。对于$\Delta\ge6$，值$\Delta+1$已知，因为仙人掌图是外平面图，因此无$K_{2,3}$-子式（Hetherington-Woodall；Agnarsson-Halldorsson）。我们的贡献在于小度情形。对于次立方仙人掌图，我们在普通和列表两种设置下都获得了完整分类，其中5-圈是唯一障碍；列表陈述没有先前的类比，是列表平方染色猜想的一个真正正例，该猜想在一般情况下是错误的。单个消去序统一处理所有$\Delta\ge4$，并特别解决了超类界留下为$\Delta+2$的$\Delta\in\{4,5\}$两种情形。数字5被证明是一个穿着三重伪装的障碍：$C_5^2=K_5$，$\{3,4\}$的弗罗贝尼乌斯数，以及一个退化的$K_4$列表染色实例。
英文摘要
The square $G^2$ of a graph joins two vertices at distance at most two; a proper coloring of $G^2$ is a 2-distance coloring of $G$. For a cactus $G$ (every edge on at most one cycle) we determine both the 2-distance chromatic number $χ(G^2)$ and the choice number $\mathrm{ch}(G^2)$ exactly: they are always equal, and the common value is $Δ+1$ if $Δ\ge4$, is 4 if $Δ=3$ and $G$ has no block equal to $C_5$, is 5 if $Δ=3$ and $G$ has a $C_5$ block, and is the classical value if $Δ\le2$. For $Δ\ge6$ the value $Δ+1$ is already known, since cacti are outerplanar and hence $K_{2,3}$-minor-free (Hetherington-Woodall; Agnarsson-Halldorsson). Our contribution is the small-degree regime. For subcubic cacti we obtain a complete classification in both the ordinary and list settings, with the 5-cycle as the unique obstruction; the list statement has no prior analogue and is a genuine positive instance of the List Square Coloring Conjecture, which is false in general. A single elimination order then handles all $Δ\ge4$ uniformly and, in particular, settles the two cases $Δ\in\{4,5\}$ that the superclass bounds leave at $Δ+2$. The number 5 turns out to be one obstruction wearing three disguises: $C_5^2=K_5$, the Frobenius number of $\{3,4\}$, and a degenerate $K_4$ list-coloring instance.
发表机构
- Sardar Vallabhbhai National Institute of Technology（萨达尔·瓦拉巴伊国家技术学院）
机构由 AI 辅助整理，请以论文原文为准。
