关于$S$-packing全染色
On $S$-packing total colorings
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中文总结 AI 辅助
本文引入$S$-packing全染色概念，推广packing全染色，建立一般上下界，刻画色数为1、2、3的图，并研究完全二部图及路径和圈的$S$-packing全色数。
中文摘要 AI 辅助
在本文中，我们通过引入一个称为$S$-packing全染色的新概念，推广了packing全染色的概念。对于图$G$和正整数的一个非递减序列$S=(a_1,a_2,\ldots)$，$G$的一个$S$-packing全染色是一个映射$c: V(G)\cup E(G)\rightarrow \{1,2,\ldots\}$，使得对于任意两个不同的元素$A,B\in V(G)\cup E(G)$，若$c(A)=c(B)=i$，则$A$和$B$之间的距离至少为$a_i+1$。使得$G$允许使用$k$种颜色的$S$-packing全染色的最小整数$k$称为$G$的$S$-packing全色数，记为$\chi_S^{''}(G)$。对于任意序列$S$，我们建立了$\chi_S^{''}(G)$的一般下界和上界，并刻画了所有满足$\chi_S^{''}(G)\in\{1,2,3\}$的图$G$。此外，我们研究了完全二部图以及无限和有限路径与圈的$S$-packing全色数。
英文摘要
In this paper, we generalize the concept of packing total coloring by introducing a new concept called the $S$-packing total coloring. For a graph $G$ and a non-decreasing sequence $S=(a_1,a_2,\ldots)$ of positive integers, an $S$-packing total coloring of $G$ is a mapping $c: V(G)\cup E(G)\rightarrow \{1,2,\ldots\}$ such that for any two distinct elements $A,B\in V(G)\cup E(G)$ with $c(A)=c(B)=i$, the distance between $A$ and $B$ is at least $a_i+1$. The smallest integer $k$ such that $G$ admits an $S$-packing total coloring using $k$ colors is called the $S$-packing total chromatic number of $G$, denoted by $χ_S^{''}(G)$. For any sequence $S$, we establish general lower and upper bounds for $χ_S^{''}(G)$, and characterize all graphs $G$ with $χ_S^{''}(G)\in\{1,2,3\}$. Furthermore, we investigate $S$-packing total chromatic numbers of complete bipartite graphs, as well as infinite and finite paths and cycles.
