L(2, 1)-labeling of the Cartesian and strong product of two directed cycles

The frequency assignment problem (FAP) is the assignment of frequencies to television and radio transmitters subject to restrictions imposed by the distance between transmitters. One of the graph theoretical models of FAP which is well elaborated is the concept of distance constrained labeling of graphs. Let \begin{document}$G = (V, E)$\end{document} be a graph. For two vertices \begin{document}$u$\end{document} and \begin{document}$v$\end{document} of \begin{document}$G$\end{document} , we denote \begin{document}$d(u, v)$\end{document} the distance between \begin{document}$u$\end{document} and \begin{document}$v$\end{document} . An \begin{document}$L(2, 1)$\end{document} -labeling for \begin{document}$G$\end{document} is a function \begin{document}$f: V → \{0, 1, ···\}$\end{document} such that \begin{document}$|f(u)-f(v)| ≥ 1$\end{document} if \begin{document}$d(u, v) = 2$\end{document} and \begin{document}$|f(u)-f(v)| ≥ 2$\end{document} if \begin{document}$d(u, v) = 1$\end{document} . The span of \begin{document}$f$\end{document} is the difference between the largest and the smallest number of \begin{document}$f(V)$\end{document} . The \begin{document}$λ$\end{document} -number for \begin{document}$G$\end{document} , denoted by \begin{document}$λ(G)$\end{document} , is the minimum span over all \begin{document}$L(2, 1)$\end{document} -labelings of \begin{document}$G$\end{document} . In this paper, we study the \begin{document}$λ$\end{document} -number of the Cartesian and strong product of two directed cycles. We show that for \begin{document}$m, n ≥ 4$\end{document} the \begin{document}$λ$\end{document} -number of \begin{document}$\overrightarrow{C_m} \Box \overrightarrow{C_n}$\end{document} is between 4 and 5. We also establish the \begin{document}$λ$\end{document} -number of \begin{document}$\overrightarrow{{{C}_{m}}}\boxtimes \overrightarrow{{{C}_{n}}}$\end{document} for \begin{document}$m ≤ 10$\end{document} and prove that the \begin{document}$λ$\end{document} -number of the strong product of cycles \begin{document}$\overrightarrow{{{C}_{m}}}\boxtimes \overrightarrow{{{C}_{n}}}$\end{document} is between 6 and 8 for \begin{document}$m, n ≥ 48$\end{document} .


(Communicated by Zhipeng Cai)
Abstract. The frequency assignment problem (FAP) is the assignment of frequencies to television and radio transmitters subject to restrictions imposed by the distance between transmitters. One of the graph theoretical models of FAP which is well elaborated is the concept of distance constrained labeling of graphs. Let G = (V, E) be a graph. For two vertices u and v of G, we denote d(u, v) the distance between u and v. An L(2, 1)-labeling for G is a function f : V → {0, 1, · · · } such that |f (u) − f (v)| ≥ 1 if d(u, v) = 2 and |f (u) − f (v)| ≥ 2 if d(u, v) = 1. The span of f is the difference between the largest and the smallest number of f (V ). The λ-number for G, denoted by λ(G), is the minimum span over all L(2, 1)-labelings of G. In this paper, we study the λ-number of the Cartesian and strong product of two directed cycles. We show that for m, n ≥ 4 the λ-number of −→ Cm2 − → Cn is between 4 and 5. We also establish the λ-number of − → C m − → C n for m ≤ 10 and prove that the λ-number of the strong product of cycles − → C m − → C n is between 6 and 8 for m, n ≥ 48.
1. Introduction. The Frequency Assignment Problem, which is the assignment of frequencies to television and radio transmitters subject to restrictions imposed by the distance between transmitters, asks for assigning frequencies to transmitters in a broadcasting network with the aim of avoiding undesired interference. This problem was first formulated as a graph coloring problem by Hale [15] under the name T -coloring. Later, a variation of the channel assignment problem was proposed in which "close" transmitters must receive different channels and "very close" transmitters must receive channels at least two apart. The problem was modeled by one For directed graphs D 1 = (V 1 , A 1 ) and D 2 = (V 2 , A 2 ), the strong product D 1 D 2 and the Cartesian product D 1 2D 2 are defined analogous to that of undirected graphs: V (D 1 D 2 ) = V (D 1 2D 2 ) = V 1 × V 2 , A(D 1 D 2 ) = {{(a, x), (b, y)}: {a, b} ∈ A 1 and x = y, or {x, y} ∈ A 2 and a = b}, and E(D 1 D 2 ) = E(D 1 2D 2 ) ∪ {{(a, x), (b, y)}: {a, b} ∈ A 1 and{x, y} ∈ A 2 }.
As an example, Figure 1 shows the Cartesian product of two directed P 6 and the Cartesian product of two directed C 6 . The Cartesian and strong product have a number of applications in engineering, computer science and related disciplines. They provide a setting in which to analyze many existing networks as well as to construct new and interesting networks [16,17,19,22,26]. Among various graphs products, the products which contains path and cycles have proved to be one of the most important [16,17,19].
The frequency assignments in which frequency inference has direction have also attracted attention in the literature [10]. They are modeled by digraphs, including ditrees [8], planar graphs [4,23], graphs with distance two [9]. In this paper, we focus on the L(2, 1)-labeling problem of the strong and Cartesian product of path and cycle, which are two of the most common and important graph products. The main result of this paper is to provide bounds for the λ-number of − → C m 2 − → C n for m, n ≥ 4 and bounds for λ-number in the infinite families of − → C m − → C n . We prove that the λ-number of the strong product of cycles − → C m − → C n is between 6 and 8 for m, n ≥ 48, and moreover, we obtain the λ-number of − → C m − → C n for m ≤ 10.
2. Preliminaries. We shall need the following well known lemma.
Let G be a graph. A function f from V (G) onto the set {0, 1, . . . , k} is called a k-labeling. If a k-labeling f of G is an L(2, 1)-labeling of G, then f is a k-L(2, 1)labeling of G.
Let f be a k-labeling of the strong or Cartesian product of two graphs. For the sake of brevity, we denote by f (x, y) the value of f (u) for u = (x, y) such that We also write f i for f i,1 . The following lemma plays an important role in the sequel.
. The function f is defined as follows: It can be seen that f is a k-L(2, 1)-labeling of − → C m+(t−1)p 2 − → C n , and the result holds.
Given two integers r and s, let S(r, s) denote the set of all nonnegative integer combinations of r and s: S(r, s) = {αr + βs : α, β ∈ Z + } Sylvester in [25] provide the following result which is useful to provide L(2, 1)labelings for infinite cases: Lemma 3. If r, s > 1 are relatively prime integers, then t ∈ S(r, s) for all t ≥ (s − 1)(r − 1).
If f is a k-L(2, 1)-labeling of a graph or digraph G, then the mirror labeling of f , denoted byf , is function from {0, 1, . . . , k} to the vertices of G such that for every It is straightforward to see Lemma 4. If f is a k-L(2, 1)-labeling of a graph or digraph G, thenf is also a k-L(2, 1)-labeling of G.
Let D 1 be a digraph with the vertex set V (D 1 ) = {x, z, w, u, v} and the set of , then the span of f is at least four.
Proof. Since u and v are at distance two from z, we have f (z) = f (v) and f (z) = f (u). Moreover, since w is adjacent to all other four vertices, it can be verified that the span of f is at least four.
Since we settled both cases, the assertion follows.
We define a digraph D n,k as follows. The vertices of D n,k are the k-L(2, 1)labelings of − → P 2 − → C n . Let u = u 1 u 2 be a vertex of D n,k . Then u 1 and u 2 represent the k-L(2, 1)-labeling of − → P 2 − → C n restricted to the first and second copy of − → C n , respectively.
Let u and v be two vertices of D n,k . Denote by uv the labeling of − → P 3 − → C n obtained by applying u 2 and v 1 to the consecutive copies of − → C n . (Note that uv is not always a k-L(2, 1)-labeling of − → P 3 − → C n ). We make an arc from u to v in D n,k if and only if, the following two conditions are fulfilled: (i) u 2 equals v 1 and (ii) uv is a k-L(2, 1)-labeling of − → P 3 − → C n . We now have the following result that follows from the results presented in [18]. In order to find closed walks in D n,k , we first try to enumerate all cycles in the graph. Directed cycles of D n,k can be found by the breadth first search or depth first search procedure if the graph of interest is not too large. As an alternative, recall that the number of distinct closed walks of length p in a digraph D can be computed via the p-th power of the adjacency matrix of D. In particular, if A is the adjacency matrix of D, then the entry (u, u) of A p equals the number of distinct closed walks of length p through u in D.
Proof. (i) The graph D 3,9 with 1800 vertices and the largest out degree 5 is created. Matrix multiplication is applied to confirm that there exists no closed walk of length in S. For the adjacency matrix A of D 3,9 we obtain that u = 41 and P = 3 such that ∀k ≥ u, A k = A u+((k−u) mod (P )) . Moreover, we find that there exist a closed walk for any length n for n ≤ 40 and n / ∈ S in D 3,9 . This assertion completes the proof of this case.
(ii) The graph D 4,8 with 2664 vertices and the largest out degree 4 is created. Matrix multiplication is applied to confirm that there exists no closed walk of length from S. For the adjacency matrix A of D 4,8 we obtain that u = 89 and P = 1 such that ∀k ≥ u, A k = A u+((k−u) mod (P )) . Moreover, we find that there exists a closed walk of any length n for n ≤ 89 and n / ∈ S in D 4,8 . This assertion completes the proof of this case.
8 10200 10 ? D 5,8 contains no closed walk of length from {6, 7, 12}, thus λ( − → C 5 − → C n ) ≥ 9 for n ∈ {6, 7, 12}.   .Column max{d + } denotes the largest out degree in D i,k , while column cycle lengths describes the set of cycles' lengths in D i,k (obtained by the depth first search procedure). If the value of the entry in cycle lengths column is denoted by ?, it means that we have failed to obtain the set by the depth first search procedure and that we have used matrix multiplication instead.

5.
Conclusion. The frequency assignment problem for wireless networks is to assign a channel to each radio transmitter so that close transmitters are received channels, so as to avoid interference. This situation can be modeled by a graph whose vertices are the radio transmitters, and the adjacency indicate possible interference. Motivated by this problem, we studied the λ-number of the Cartesian and strong product of two directed cycles. We show that for m, n ≥ 4 the λ-number of − → C m 2 − → C n is between 4 and 5. The second part of the paper is devoted to the λnumber of the strong product of two directed cycles. We prove that the λ( − → C m − → C n ) is between 6 and 8 for m, n ≥ 48. Moreoveor, we obtain the λ-numbers of − → C m − → C n for every m ≤ 10.