Abstract
An -labeling of a graph is a function from the vertex set to the set of nonnegative integers such that the labels on adjacent vertices differ by at least two and the labels on vertices at distance two differ by at least one. The span of is the difference between the largest and the smallest numbers in . The -number of , denoted by , is the minimum span over all -labelings of . We consider the -number of and for the -number of . We determine -numbers of graphs of interest with the exception of a finite number of graphs and we improve the bounds on the -number of , and .
1. Introduction
The Frequency Assignment Problem (FAP) requires assigning frequencies to transmitters in a wireless network. In a broadcasting network, each transmitter is assigned a frequency channel for its transmissions. Two transmissions can interfere if their channels are too close. That means that even if two transmitters use different channels, there still may be interference if the two transmitters are located close to each other [1, 2].
The spectrum of frequencies gets more and more scarce because of increasing demands, both civil and military. Thus the task is to minimize the span of frequencies while avoiding interference. One of the graph-theoretical models of FAP which is well elaborated is the concept of distance-constrained labeling of graphs [1]. Many variants of this concept have been proposed; however, the -labeling problem where adjacent vertices must be assigned colors of distance at least two apart and vertices of distance two must be assigned different colors has attracted the most of interest [3, 4].
An -labeling of a graph is a function from the vertex set to the set of nonnegative integers (called labels or colors) such that for any two vertices and
A -labeling is a -labeling of such that . An optimal -labeling of is a -labeling with smallest possible. The largest label used by an optimal -labeling is called the -number of and denoted by .
There is a number of studies on the algorithms for -labeling problem [1, 5, 6]. It is known to be NP-hard for general graphs [4]. Even for some relatively simple families of graphs such as planar graphs, bipartite graphs, chordal graphs [5], and graphs of treewidth two [7], the problem is also NP-hard.
Product graphs are considered in order to gain global information from the factor graphs [8]. Many interesting wireless networks are based on product graphs with simple factors, such as paths and cycles. In particular, any square grid (resp., torus) is the Cartesian product of two paths (resp., cycles) and any octagonal grid (resp., torus) is the strong product of two paths (resp., cycles) [9]. For the Cartesian product of these factors the numbers have been completely determined [10–12], while for the strong and the direct product only partial results have been found [13–15].
The paper is organized as follows. In Section 2, we give definitions and concepts needed in this paper. We also report on the known results for the numbers of the graphs of interest. In Section 3, two main computer search methods applied in the paper are described: the dynamic algorithm and the SAT reduction. Finally, in Section 4, we present the results on the -number of and the -number of .
2. Preliminaries and Previous Results
For a graph , and are the sets of vertices and edges of , respectively. A directed graph consists of vertices together with a set of arcs . We write also to stand for the vertex set of the graph . In this paper, only directed and undirected graphs without multiple edges or loops are considered.
The strong product of graphs and is the graph with vertex set and whenever and , or and , or and . The strong product is commutative and associative, having the trivial graph as a unit (cf. [8]). The subgraph of induced by is isomorphic to . It is called an -fiber and denoted by .
The path is the graph whose vertices are and for which two vertices are adjacent precisely if their difference is . For an integer , the cycle of length is the graph whose vertices are and whose edges are the pairs , , where the arithmetic is done modulo . Note that the strong product , depicted in Figure 1, can be regarded as a graph composed of six copies of (denoted by ) or a graph composed of 13 copies of (denoted by ).
A walk in a directed graph is a sequence of (not necessarily distinct) vertices , such that for . If , we say it is a closed walk.
If is a path (resp., walk), then its length is its number of edges (resp., arcs).
The following simple lemma is well known.
Lemma 1. If is a subgraph of , then .
Let denote a --labeling of . We denote by the restriction of to , and . Note that is isomorphic to ; that is, is the subgraph of induced by . We will also write for .
The following lemma provides an upper bound for the -number of the strong product of a graph with a cycle.
Lemma 2. Let and be a -labeling of . If is a -labeling of , then .
Proof. Let be a function from onto the set and the restriction of to . The function is defined as follows: It is not difficult to see that is a -labeling of .
Given two integers and , let denote the set of all nonnegative integer combinations of and :
We will need the result of Sylvester [16].
Lemma 3. If are relatively prime integers, then for all .
Some partial results on the -number for the strong products of two cycles are given in [15].
Theorem 4. Let . Then
Theorem 5. Let . Then
Theorem 6. If and , then .
For the strong product of more than two cycles the following result presented in [14] is known.
Theorem 7. If and are each multiple of , then .
3. Computer Search
3.1. Dynamic Algorithm
The idea is introduced in [10] in a more general framework and later used several times, for example, [13, 15]. In order to make the paper self-contained we first describe its basic definitions and results.
We define a digraph as follows. Its vertices are the -labelings of . Let be a vertex of . Then and represent the -labeling of restricted to the first and second copies of , respectively.
Let and be two vertices of . Then denotes the labeling of obtained by applying , and to the consecutive copies of . (Note that is not always a -labeling of .) We make an arc from to in if and only if the following two conditions are fulfilled:
equals ; is a -labeling of .
Analogously, we define a digraph with the vertex set composed by -labelings of . In other words, if is a vertex of , then and represent the -labeling of restricted to the first and second copies of , respectively. The set of arcs of is formed analogously as the set of arcs of .
Figure 2(a) shows two vertices of denoted by and . We can see that the labeling of the second copy of in equals the labeling of the first copy of in . Moreover, the labeling of and the labeling of the second copy of in induce a -labeling of . It follows that admits an arc from to .
(a)
(b)
The next theorem follows from the results presented in [10].
Theorem 8. (resp., ) admits a -labeling if and only if (resp. ) contains a closed directed walk of length .
The dynamic algorithm first generates all -labelings of which are the vertices of . Since a main building block is usually relatively small, a simple method, for example, backtracking, can be applied for this step. In the next step, the set of edges of has to be generated. The procedure for this step is described in [15]. The described algorithm however has the time complexity , where denotes the number of vertices in . Note that can be very large even for and of a moderate size. Some examples for and of interest are , , and . The complexity of the algorithm does not allow a computation of in a reasonable time for these cases. We have therefore improved this method as described in the sequel.
Let denote the set of all --labelings of . If is an element of , then , , denote the restriction of to the first, second, and third copies of in ; respectively. Note that contains symmetric labelings of , that is, if , then exists, such that , , and .
We now define the digraph as follows. Let denote the set of -labelings of obtained from in the following way: belongs to if and only if there exist such that , , and , .
Figure 2(b) shows two vertices of denoted by and . We can see that the labelings of the first and the second copies of in equal the third and the second copies of in , respectively. It follows that possesses the vertex comprising these two labelings.
Let . We make an arc from to if and only if and belongs to .
Note that analogous as above we can improve the method for . The graph obtained with this procedure (a subgraph of ) will be denoted by in the sequel.
For a vertex of a directed graph , the number of inward (resp., outward) directed arcs from in is called an (resp., ) and denoted by (resp., ).
We obtain the main result of this section.
Theorem 9. (resp., ) admits a -labeling if and only if (resp., ) contains a closed directed walk of length .
Proof. It is easy to see that and . From the definition of for it follows that and .
Suppose now for that and . Since contains all -labelings of and , there has to be a vertex such that and . Moreover, since , there has to be a vertex such that and . It follows that belongs to . We have therefore proven that if and only if and .
Analogously as above we can show that is an arc in if and only if is an arc in . In other words, we can show that is isomorphic to subgraph of induced by . It follows that contains a closed directed walk of length if and only if contains a closed directed walk of the same size and the proof for is settled.
Since the proof for is analogous, the proof of the theorem is complete.
The algorithm for generating the graph is depicted in Algorithm 1 (Procedure CREATE GRAPH).
| ||||||||||||||||||||
Note that the number of vertices of can be much smaller than in . Some examples for and of interest are: , , and . However, this reduction is not the only positive effect of the new approach. It is also of a great importance that the running time of CREATE GRAPH is , where , the number of vertices in . In order to see this, note that the running time of an efficient sorting algorithm is also within this time bound. Moreover, this is also the running time of the duration of loop, since a single search in an ordered list with elements requires time.
The final step of the approach is the search for closed direct walks in . We can find these walks by applying a matrix multiplication of the adjacency matrix of or a breadth (depth) first search in . Since graphs are relatively sparse for and of interest, the later approach has been applied in order to compute the results of this paper.
3.2. SAT Reduction for -Labeling
The approach is proposed in [17] for the distance-constrained labeling problem. Here we present this method adapted for -labeling.
Let be a graph and a positive integer. For every and every introduce an atom . Intuitively, this atom shows that the vertex is assigned the color . Consider the following propositional formulas:(1)for all , ;(2)for all , , ;(3)for all , , , with and or and , .
Clauses and ensure that each vertex is labeled with exactly one label. Clause guarantees that an obtained labeling is a -labeling of . Therefore, the above propositional formulas transform an -labeling problem into a propositional satisfiability test (SAT). We can see that an obtained SAT instance is satisfiable if and only if admits a -labeling.
4. Results
4.1. SAT Reduction
We solve the SAT instances transformed from -labeling problems described in Section 4.2 by using the software MiniSat [18]. As a result, we have obtained the -numbers of presented in Table 1 and the -numbers of presented in Table 2.
The values in Table 2 marked with denote the results already obtained in [15], while the entry with means that the corresponding value is either 13 or 14.
4.2. -Labeling of
Proposition 10. only if .
Proof. Note that . We can see that from the fact that every pair of vertices is at distance at most two. Let denote a -labeling of and its restriction to . Let also denote the set of labels used in . Since , we have and . Therefore, the restriction of to has to comprise the same set of labels as the restriction of to or, more formally, . It is straightforward to see that this equality can be satisfied in only if (mod 3).
Theorem 11. Let . Then
Proof. Note that the values for are given in Table 2. We can also show by using the SAT reduction that . Since only if , we construct below a -labeling for , , a -labeling for , , and a -labeling for , .
Let for denote the set .
Let denote a function from to and its restriction to , . Let also denote the set of labels used in . If we set for and : , then is a -labeling of for .
Let denote a function from to and its restriction to , . Let also denote the set of labels used in . If we set for for : , , and for and : , then is a -labeling of for . As an example, observe the following pattern representing a -labeling of :15 4 9 14 3 8 13 2 7 12 1 6 11 0 5 10 0 5 10 0 5 10 0 5 10 15 4 9 14 3 8 13 2 7 12 1 6 11 1 6 11 1 6 11 1 6 11 0 5 10 15 4 9 14 3 8 13 2 7 12 2 7 12 2 7 12 2 7 12 1 6 11 0 5 10 15 4 9 14 3 8 13 3 8 13 3 8 13 3 8 13 2 7 12 1 6 11 0 5 10 15 4 9 14 4 9 14 4 9 14
Let denote a function from to and its restriction to , . Let also denote the set of labels used in . If we set for for : and for and : , then is a -labeling of for . This assertion concludes the proof.
The following results partially depend on comprehensive constructions which provide labelings of interest. These constructions are mostly not included in this paper and can be obtained by the authors.
Proposition 12. only if .
Proof. The graph with 12336 vertices and the largest outdegree six has been computed. Since breadth first search algorithm has found only cycles of length four, Theorem 9 yields the proof.
Theorem 13. If , then
Proof. The results for follow from Table 2. We can also see in Table 1 that ; thus, from Lemma 1 it follows that . Moreover, Proposition 12 says that only if . In order to see that for other of interest, see as an example a -labeling of depicted in Figure 1.
From Lemma 2 it follows that if . Analogously, we have found -labelings of , , , , , , , , , , , and . Any of these labelings restricted to the first 13 copies of induces a -labeling of . From Lemma 2 it follows that for , for , for , for , for , for , for , for , for , for , for , and for .
Since we have also found -labelings of , , , , , and , we conclude that for and the proof is complete.
Theorem 14. If , then
Proof. The results for follow from Table 2. We have also established the results for by solving the SAT instances transformed from the corresponding -labeling problems. Since as we can see in Table 2, it follows by Lemma 1 that for . In order to see that for other of interest, see as an example a -labeling of depicted in Figure 3. This labeling restricted to the first ten copies of induces a -labeling of . From Lemma 2 it follows that , , and , .
Analogously, we have found -labelings of , , , , , , , and . Any of these labelings restricted to the first ten copies of is a -labeling of . From Lemma 2 it follows that for , for , for , for , for , for , for , for , and for .
Since we have also found -labelings of , , , and , it follows that for all graphs of interest and the proof is complete.
Proposition 15. only if .
Proof. The graph with 8157632 vertices and the largest outdegree 8 has been computed. Since breadth first search algorithm has found only cycles of length, six, twelve, and twenty-four, Theorem 9 yields the proof.
Theorem 16. If , then
Proof. Since (see Table 2), it follows from Lemma 1 that . Proposition 15 says that only if , while the results for follow from Table 2. Figure 4 shows a -labeling of . This labeling restricted to the first nine copies of induces a -labeling of . From Lemma 2 it follows that , , and , .
Moreover, we have found -labelings of , , , , , , and . Any of these labelings restricted to the first nine copies of is a -labeling of . From Lemma 2 it follows that for , for , for , for , for , for , and for .
Since we have also found -labelings of , , , , , , , , and , we establish the desired upper bound for all graphs of interest and the proof is complete.
Theorem 17. If , then
Proof. Since (see Table 2), it follows from Lemma 1 that . The results for follow from Table 2. We have also established by solving the SAT instances transformed from the corresponding -labeling problems that is either 13 or 14, while for the value of is either 12 or 13. Figure 5 shows a -labeling of . This labeling restricted to first 12 copies of induces a -labeling of . From Lemma 2 it follows that , , and , .
We have also found -labelings of , , , , , , , , , and . Any of these labelings restricted to first 12 copies of induces a -labeling of . From Lemma 2 it follows that for , for , for , for , for , for , for , for , for , and for .
Since we have also found -labelings of , , , , , , , , , , and , we establish that for all of interest and the proof is complete.
Theorem 18. If , then
Proof. We can see in Table 2 that hence, it follows by Lemma 1 that . The results for are given in Table 2.
The following pattern represents a -labeling of , while the leftmost 12 columns of the pattern represents a -labeling of :8 3 1 12 8 5 7 2 0 11 9 6 8 3 1 12 10 7 4 2 0 11 9 6 8 3 1 12 10 5 7 4 2 0 11 9 6 0 11 9 6 3 1 12 10 5 7 4 2 0 11 9 6 3 1 12 10 5 7 4 2 0 11 9 6 8 3 1 12 10 5 7 4 2 7 4 2 0 11 9 6 8 3 1 12 10 5 7 2 0 11 9 6 8 3 1 12 10 5 7 4 2 0 11 9 6 8 3 1 12 10 1 12 10 5 7 4 2 0 11 9 6 3 1 12 10 5 7 4 2 0 11 9 6 8 3 1 12 10 5 7 4 2 0 11 9 6 3 9 6 8 3 1 12 10 5 7 2 0 11 9 6 8 3 1 12 10 5 7 4 2 0 11 9 6 8 3 1 12 10 5 7 2 0 11 4 2 0 11 9 6 3 1 12 10 5 7 4 2 0 11 9 6 8 3 1 12 10 5 7 4 2 0 11 9 6 3 1 12 10 5 7 12 10 5 7 4 0 11 9 4 8 3 1 12 10 5 7 4 2 0 11 9 6 8 3 1 12 10 5 7 2 0 11 9 6 8 3 1 6 8 3 1 12 8 5 7 2 0 11 9 6 8 3 1 12 10 5 7 4 2 0 11 6 8 3 1 12 10 5 7 4 2 0 11 9 2 0 11 9 6 3 1 12 10 5 7 4 2 0 11 9 6 8 3 1 12 10 7 4 2 0 11 9 6 8 3 1 12 10 5 7 4 10 5 7 4 0 11 9 4 8 3 1 12 10 5 7 4 2 0 11 6 8 3 1 12 10 5 7 4 2 0 11 9 6 8 3 1 12
By Lemma 2, we have for integers and . Finally, thanks to Lemma 3, we get for .
We have found -labelings of for and we can construct -labelings of for as follows. We have found -labelings of , , , , , , and . Any of these labelings restricted to the first eight copies of is a -labeling of . From Lemma 2 it follows that for , for , for , for , for , for , for , and for . These observations complete the proof.
Proposition 19. only if .
Proof. The graph with 380 vertices and the largest outdegree 2 has been computed. Since breadth first search algorithm has found only cycles of length length 11, Theorem 9 yields the proof.
Theorem 20. If , then
Proof. For the numbers are obtained by using the SAT reduction as depicted in Table 1. Since , from Lemma 1 it follows that , while from Proposition 19 it follows that if (mod 11).
The result for can be obtained by the fact that and by Lemma 2.
Figure 6 represents an -labeling of , where the leftmost 11 columns of the figure represent an -labeling of . By Lemma 2, we have for integers and . Finally, thanks to Lemma 3, we get for .
In order to find the general upper bound, we present the constructions showing that for . In particular, Figure 7 shows a -labeling of . This labeling restricted to the first 12 copies of induces a -labeling of . From Lemma 2 it follows that , , and , .
Analogously, we have found -labelings of , , , , , , , , , and . Any of these labelings restricted to the first 12 copies of is a -labeling of .
From Lemma 2 it follows that for , for , for , for , for , for , for , for , for , and for .
Since we have also found -labelings of , , , , , , , and , we establish that for all and the proof is complete.
4.3. -Numbers of
Proposition 21. If , then
Proof. For the results are obtained by solving the SAT instances transformed from the corresponding -labeling problems (see Table 1).
The graph with 9080 vertices and the largest outdegree 16 has been created in order to find -labelings in . Matrix multiplication has been applied in order to find closed directed walks in the graph. The algorithm has found no closed directed walk of length from the set . It follows that for any . The upper bounds for follow from the labelings depicted in Figure 8.
We have found -labelings of , , , , , , , , , and . Any of these labelings restricted to the first 11 copies of is a -labeling of . From Lemma 2 it follows that for , for , for , for , for , for , for , for , for , for , and for .
Since we have also found -labelings of , , , , , , , , , , , , , , , , and , we establish that for all and the proof is complete.
(a)
(b)
(c)
(d)
(e)
(f)
Proposition 22. If , then
Proof. For the result is obtained by solving the SAT instance transformed from the corresponding -labeling problem. For the result follows from Lemma 2 and from the fact that .
In order to find -labelings in , the graph with 16792 vertices and the largest outdegree 3 has been created. Since breadth first search algorithm has found only cycles of length 11, the upper bound follows.
For all and we can construct -labelings of as described below. We have found -labelings of , , , , and . Any of these labelings restricted to the first six copies of is an -labeling of . From Lemma 2 it follows that for , for , for , for , and for . These conclusions complete the proof.
Corollary 23. Let .(i)If , then ,(ii)If or , then .
From Theorem 6 now we have the following
Corollary 24. If and , then
Proposition 25. If , then
Proof. For the results follow from Table 1. For the result follows from Lemma 2 and from the fact that .
Figure 9 represents an -labeling of , where the leftmost six columns of the figure represent an -labeling of .
By Lemma 2, we have for integers and . Finally, thanks to Lemma 3, we get for .
In order to complete the proof note that from Theorem 16 and Lemma 1 it follows that for .
Proposition 26. If , then
Proof. For the results follow from Table 1. For the result follows Lemma 2 and from the fact that .
Figure 10 represents a -labeling of , where the leftmost eight columns of the figure represent an -labeling of .
By Lemma 2, we have for integers and . From Lemma 3 it follows that for .
We complete the proof by noting that from Theorem 16 and Lemma 1 it follows for .
Values in Table 1, the results from Section 4.2, Theorems 6 and 20, and Corollary 23 provide lower and upper bounds for the -number of . The results are summarized in the following.
Theorem 27. If , then
5. Conclusion
In this paper, the -labeling problem of the strong product of paths and cycles is studied. The problem derives from the more general Frequency Assignment Problem (FAP) which requires assigning frequencies to transmitters in a wireless network. It is well known that some interesting wireless networks are closely connected to the strong product of graphs. For example, an octagonal grid is the strong product of two paths and an octagonal torus is the strong product of two cycles.
By using various computational approaches, we succeed in solving the problem (except for the final number of cases) for the strong product of a path and a cycle, as well as for the the strong product of two cycles, where one of the cycles is of length at most eleven. Moreover, the obtained results enable us to improve the bounds on the -number for the strong product of two cycles, where both cycles are sufficiently long. Finding the exact -numbers for these graphs is therefore an interesting and challenging avenue of further research.
Conflict of Interests
The authors wish to confirm that there is no known conflicts of interests associated with this paper and there has been no significant financial support for this work that could have influenced its outcome. They confirm that the paper has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. They further confirm that the order of authors listed in the paper has been approved by all of us. They confirm that they have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing they confirm that they have followed the regulations of their institutions concerning intellectual property.
Acknowledgments
Aleksander Vesel is supported by the Ministry of Science of Slovenia under the Grant 0101-P-297. Zehui Shao is supported by the National Natural Science Foundation of China under the Grant 61309015.