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 $$d(A,B)=\min \{|a-b|:a\in A,b\in B\}$$ d ( A , B ) = min { | a - b | : a ∈ A , b ∈ B } . For a positive integer n, let $$S_{n}$$ S n denote the family $$ \{X:X\subseteq {\mathbb {N}} \cup \{0\},|X|=n\}$$ { X : X ⊆ N ∪ { 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 $$f:V(G)\rightarrow S_{n} $$ f : V ( G ) → S n satisfies $$d(f(u),f(v))\ge p$$ d ( f ( u ) , f ( v ) ) ≥ p if $$d_{G}(u,v)=1$$ d G ( u , v ) = 1 , and $$ d(f(u),f(v))\ge q$$ d ( f ( u ) , f ( v ) ) ≥ q if $$d_{G}(u,v)=2$$ 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 $$\max \{a:a\in \bigcup _{u\in V(G)}f(u)\}\le k$$ max { a : a ∈ ⋃ u ∈ V ( G ) f ( u ) } ≤ 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 $$\lambda _{p,q}^{n}(G)$$ λ p , q n ( G ) . When $$p=2, $$ p = 2 , $$q=1,$$ q = 1 , we use $$\lambda ^{n}(G)$$ λ n ( G ) to replace $$ \lambda _{2,1}^{n}(G)$$ λ 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 $$\lambda ^{n}(C_{m}\square P_{2})$$ λ n ( C m □ P 2 ) equals $$4n+1.$$ 4 n + 1 . Based on this, we determine the exact value of $$ \lambda ^{2}(C_{m}\square P_{2})$$ λ 2 ( C m □ P 2 ) (except for $$m=5,6$$ m = 5 , 6 and 9) and $$\lambda ^{3}(C_{m}\square P_{2})$$ λ 3 ( C m □ P 2 ) (except for $$m=5,6,9,10,13$$ m = 5 , 6 , 9 , 10 , 13 and 17). We also give bounds for $$\lambda ^{n}(C_{m}\square P_{k})$$ λ n ( C m □ P k ) when n, m satisfy certain conditions, and from this, we obtain the exact value of $$\lambda ^{2}(P_{m}\square P_{k})$$ λ 2 ( P m □ P k ) (except for the case $$P_{4}\square P_{3}$$ P 4 □ P 3 ).