路径和循环的笛卡尔乘积的 n 折 L(2, 1)标记

IF 0.9 4区 数学 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Fei-Huang Chang, Ma-Lian Chia, Shih-Ang Jiang, David Kuo, Jing-Ho Yan
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引用次数: 0

摘要

对于两个非负整数集合 A 和 B,这两个集合之间的距离(用 d(A,B)表示)由 \(d(A,B)=\min \{|a-b|:a\in A,b\in B\}\) 定义。对于正整数 n,让 \(S_{n}\) 表示族 \( \{X:X\subseteq {\mathbb {N}} \cup \{0\},|X|=n\}\).给定一个图 G 和正整数 n、p 和 q,G 的 n 折 L(p, q)标记是一个函数 \(f:V(G)\rightarrow S_{n}) 满足 \(f:V(G)\rightarrow S_{n}).\如果(d_{G}(u,v)=1),则满足(d(f(u),f(v));如果(d_{G}(u,v)=2),则满足(d(f(u),f(v))。G的n折k-L(p,q)-标签f是G的n折L(p,q)-标签,其性质是(max \{a:a\in \bigcup _{u\in V(G)}f(u)\}\le k\ )。保证 G 有 n 重 k-L(p, q)标记的最小数 k 称为 G 的 n 重 L(p, q)标记数,用 \(\lambda _{p,q}^{n}(G)\) 表示。当 \(p=2,\) \(q=1,\) 时,为了简单起见,我们用 \(\lambda ^{n}(G)\) 代替 \( \lambda _{2,1}^{n}(G)\) 。本文研究了路径与循环的笛卡尔积的 n 折 L(2, 1) -标签数。我们给出了(\lambda ^{n}(C_{m}\square P_{2})\) 等于(4n+1.\在此基础上,我们确定了\(\lambda ^{2}(C_{m}\square P_{2})\) (除了\(m=5,6\)和9)和\(\lambda ^{3}(C_{m}\square P_{2})\) (除了\(m=5,6,9,10,13\)和17)的精确值。)当 n、m 满足一定条件时,我们还给出了 \(λ ^{n}(C_{m}\square P_{k})\) 的边界,并由此得到了 \(λ ^{2}(P_{m}\square P_{k})\) 的精确值(除了 \(P_{4}\square P_{3}\) 的情况)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

n-fold L(2, 1)-labelings of Cartesian product of paths and cycles

n-fold L(2, 1)-labelings of Cartesian product of paths and cycles

For two sets of nonnegative integers A and B, the distance between these two sets, denoted by d(AB), is defined by \(d(A,B)=\min \{|a-b|:a\in A,b\in B\}\). For a positive integer n, let \(S_{n}\) denote the family \( \{X:X\subseteq {\mathbb {N}} \cup \{0\},|X|=n\}\). Given a graph G and positive integers n, p and q, an n-fold L(pq)-labeling of G is a function \(f:V(G)\rightarrow S_{n} \) satisfies \(d(f(u),f(v))\ge p\) if \(d_{G}(u,v)=1\), and \( d(f(u),f(v))\ge q\) if \(d_{G}(u,v)=2\). An n-fold k-L(pq)-labeling f of G is an n-fold L(pq)-labeling of G with the property that \(\max \{a:a\in \bigcup _{u\in V(G)}f(u)\}\le k\). The smallest number k to guarantee that G has an n-fold k-L(pq)-labeling is called the n -fold L(pq)-labeling number of G and is denoted by \(\lambda _{p,q}^{n}(G)\). When \(p=2, \) \(q=1,\) we use \(\lambda ^{n}(G)\) to replace \( \lambda _{2,1}^{n}(G)\) for simplicity. We study the n-fold L(2, 1) -labeling numbers of Cartesian product of paths and cycles in this paper. We give a necessary and sufficient condition for \(\lambda ^{n}(C_{m}\square P_{2})\) equals \(4n+1.\) Based on this, we determine the exact value of \( \lambda ^{2}(C_{m}\square P_{2})\) (except for \(m=5,6\) and 9) and \(\lambda ^{3}(C_{m}\square P_{2})\) (except for \(m=5,6,9,10,13\) and 17). We also give bounds for \(\lambda ^{n}(C_{m}\square P_{k})\) when nm satisfy certain conditions, and from this, we obtain the exact value of \(\lambda ^{2}(P_{m}\square P_{k})\) (except for the case \(P_{4}\square P_{3}\)).

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来源期刊
Journal of Combinatorial Optimization
Journal of Combinatorial Optimization 数学-计算机:跨学科应用
CiteScore
2.00
自引率
10.00%
发文量
83
审稿时长
6 months
期刊介绍: The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering. The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.
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