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

Abstract

For two sets of nonnegative integers A and B, the distance between these two sets, denoted by d(A, B), 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(p, q)-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(p, q)-labeling f of G is an n-fold L(p, q)-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(p, q)-labeling is called the n -fold L(p, q)-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 n, m satisfy certain conditions, and from this, we obtain the exact value of $\lambda ^{2}(P_{m}\square P_{k})$ (except for the case $P_{4}\square P_{3}$).