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(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 \(P_{4}\square P_{3}\)).