乘积图的算术亲切标号
On Arithmetic Cordial Labeling of Product Graphs
中文总结 AI 辅助
本文研究加法运算下 corona、字典积、笛卡尔积、张量积、强积等乘积图的算术亲切标号，定义了模η的算术亲切标号并探讨其相关性质。
中文摘要 AI 辅助
设η为固定正整数，S是ℤ的子集，⋆: S×S→ℤ是二元函数，ζ_η: {ξ∈ℤ: gcd(ξ,η)=1}→{0,1}是函数。对于n阶简单图G，双射函数f: V(G)→S（|S|=n）被称为在算术结构⟨S,ζ_η,⋆⟩下模η的算术亲切标号，当由f定义的诱导函数f_η^*: E(G)→{0,1}满足：当gcd(f(a)⋆f(b),η)=1且ζ_η(f(a)⋆f(b))=1时，f_η^*(ab)=1，否则为0；且|e_{f_η^*}(0)-e_{f_η^*}(1)|≤1，其中e_{f_η^*}(i)是标签为i（i=0,1）的边数。本文在加法运算下，研究乘积图的算术亲切标号，包括 corona、lexicographic、cartesian、tensor 和 strong 乘积图。
英文摘要
Let $η$ be a fixed positive integer. Let $S$ be a subset of $\mathbb{Z}$, $\star:S\times S\to \mathbb{Z}$ be a binary function, and $ζ_η:\{ξ\in \mathbb{Z}:\gcd(ξ,η)=1\}\to \{0,1\}$ be a function. For a simple graph $G$ of order $n$, a bijective function $f:V(G)\to S$ (where $|S|=n$) is called an arithmetic cordial labeling modulo $η$ under the arithmetic structure $\langle S,ζ_η,\star\rangle$ if the induced function $f_η^*:E(G)\to \{0,1\}$, defined by $f_η^*(ab)=1$ whenever $\gcd(f(a)\star f(b),η)= 1$ and $ζ_η(f(a)\star f(b))=1$; otherwise, $f_η^*(ab)=0$, satisfies the condition $|e_{f_η^*}(0)-e_{f_η^*}(1)|\leq 1$, where $e_{f_η^*}(i)$ is the number of edges with label $i$ ($i=0,1$). In this paper, the arithmetic cordial labeling of product graphs, namely, corona, lexicographic, cartesian, tensor, and strong, is explored under the operation of addition.
