将$S$-打包染色划分为广播控制集

中文摘要 AI 辅助

给定一个图$G$和一个正整数$k$，$G$的一个打包$k$-控制染色是一个函数$f: V(G) \to \{1,\ldots,t\}$，使得(1)对于任意颜色$j \in \{1,\ldots, t\}$以及任意两个不同的顶点$u,v\in V(G)$，若$f(u)=f(v)=j$，则$d_G(u,v)>j$，且(2)$V(G)$可划分为$k$个集合$A_1,\ldots,A_k$，使得对于任意$v\in V(G)$和任意$i\in\{1,\ldots,k\}$，存在一个顶点$w\in A_i$满足$d_G(v,w)\le f(w)$。使得$G$允许使用$\{1,\ldots,t\}$中颜色的打包$k$-控制染色的最小整数$t$记为$\chi_{\rho,k}(G)$。打包$k$-控制染色定义中的条件(1)蕴含$G$是一个打包染色，因此对任意图$G$有$\chi_{\rho,k}(G)\ge \chi_\rho(G)$，其中$\chi_\rho(G)$是$G$的打包色数。另一方面，这一新概念也导致了对图的控制数的推广，由此容易看出，对于任何没有孤立顶点的图$G$，$\chi_{\rho,2}(G)=\chi_\rho(G)$成立。本文的主要结果之一是：在任何最小度至少为2的连通图$G$中，$\chi_{\rho,3}(G)=\chi_\rho(G)$成立，并且该界在两种意义下都是最优的。此外，我们提供了路径和圈上这一新不变量的若干精确值和界。我们证明，对于任意$k\in\{3,4,5\}$，只要$n\ge 8$，就有$\chi_{\rho,k}(P_n)=k$；相反地，对于任意$n\ge k\ge 12$，$\chi_{\rho,k}(P_n)>k$。我们还证明了当$k\in \{3,4,5\}$时双向无限路径$P_\infty$的$\chi_{\rho,k}(P_\infty)$的精确值，以及当$k\in\{3,4\}$时所有圈$C_n$的$\chi_{\rho,k}(C_n)$的精确值。我们还考虑了一个更一般的框架，即$S$-打包$k$-控制染色，其中$S$是任意非负整数序列，并在这一背景下给出了一些基本结果。

英文摘要

Given a graph $G$ and a positive integer $k$, a packing $k$-domatic coloring of $G$ is a function $f: V(G) \to \{1,\ldots,t\}$ such that (1) for any color $j \in \{1,\ldots, t\}$ and any two distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=j$ we have $d_G(u,v)>j$, and (2) $V(G)$ admits a partition into $k$ sets $A_1,\ldots,A_k$ such that for any $v\in V(G)$ and any $i\in\{1,\ldots,k\}$ there exists a vertex $w\in A_i$ such that $d_G(v,w)\le f(w)$. The minimum integer $t$ such that $G$ admits a packing $k$-domatic coloring of $G$ using colors in $\{1,\ldots,t\}$ is denoted by $χ_{ρ,k}(G)$. The condition (1) in the definition of a packing $k$-domatic coloring implies that $G$ is a packing coloring, hence $χ_{ρ,k}(G)\ge χ_ρ(G)$ holds for any graph $G$, where $χ_ρ(G)$ is the packing chromatic number of $G$. On the other hand, the new concept also leads to a generalization of the domatic number of a graph due to which one can easily see that $χ_{ρ,2}(G)=χ_ρ(G)$ holds for any graph with no isolated vertices. One of the main result in this paper is that $χ_{ρ,3}(G)=χ_ρ(G)$ holds in any connected graph $G$ with minimum degree at least $2$, and the bound is best possible in two different senses. In addition, we provide several exact values and bounds on the new invariant in paths and cycles. We prove that $χ_{ρ,k}(P_n)=k$ for any $k\in\{3,4,5\}$ as soon as $n\ge 8$, and, in contrast, $χ_{ρ,k}(P_n)>k$ for any $n\ge k\ge 12$. We also prove the exact values of $χ_{ρ,k}(P_\infty)$ when $k\in \{3,4,5\}$ for the two-way infinite path $P_\infty$, and exact values of $χ_{ρ,k}(C_n)$ when $k\in\{3,4\}$ for all cycles $C_n$. We also consider a general framework of $S$-packing $k$-domatic colorings, where $S$ is an arbitrary sequence of non-negative integers, and present some basic results in this context.

发表机构

• Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）
• Institute of Mathematics, Physics and Mechanics, Ljubljana（卢布尔雅那数学、物理与力学研究所）
• Faculty of Education, University of Maribor（马里博尔大学教育学院）

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26 pages, 1 figure