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Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 163740, 10 pages
A Theory of Cartesian Product and Factorization of Circulant Graphs
Department of Mathematics, St. Jude's College, Thoothoor, Tamil Nadu 629 176, India
Received 27 May 2013; Revised 14 September 2013; Accepted 16 September 2013
Academic Editor: Wai Chee Shiu
Copyright © 2013 V. Vilfred. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We determine when the Cartesian product of two circulant graphs is also a circulant graph. This leads to a theory of factorization of circulant graphs.
Just as integers can be factored into prime numbers, there are many results on decomposition of structures throughout mathematics . The standard products—Cartesian, lexicographic, tensor, and strong—all belong to a class of products introduced by Imrich and Izbicki  and called -products . Properties of circulant graphs are extensively studied by many authors [2–19] and products of graphs have been studied for almost 50 years now. Sabidussi  proved that every (nonnull) graph is the unique product of prime graphs. Broere and Hattingh  established that the lexicographic product of two circulant graphs is again circulant. They and Sanders and George  established that this is not the case with other products. Alspach and Parsons  introduced metacirculant graphs as a generalization of circulant graphs and characterized metacirculant graphs in terms of their automorphism groups. Sanders  established that any -product of two circulant graphs is always a metacirculant graph with parameters that are easily described in terms of the product graphs and also established that any metacirculant graph with the appropriate structure is isomorphic to the -product of a pair of circulant graphs. After a graph is identified as a circulant graph, its properties can be derived easily. This paper gives a detailed study of Cartesian product and factorization of circulant graphs similar to the theory of product and factorization of natural numbers. For more details on circulant graphs, see [9, 10].
Let be a positive integer and let be a subset of . The circulant graph has vertices with adjacent to for each , subscript addition taken modulo . When discussing circulant graphs, we will often assume, without further comment, that the vertices are the corners of a regular -gon, labeled clockwise. Circulant graphs and are shown in Figures 1(a) and 1(b).
When , edge is taken as a single edge while considering the degree of a vertex, but as a double edge while counting number of edges or cycles in [3, 6–10, 13, 14, 17, 18, 21]. We generally write for and for , the null graph on vertices. Note that if a graph is circulant, then its adjacency matrix is circulant. It follows that if the first row of the adjacency matrix of a circulant graph is , then and = , .
Throughout this paper, for a set denotes where ; we consider only connected circulant graphs of finite order greater than 2 and all cycles have length at least 3, unless otherwise specified.
Let , be positive integers, and . Then, consists of a collection of disjoint cycles, ; ; . If , then there are such disjoint cycles and each has length . We say that each of these cycles has period , length , and rotation .
If , then obviously is just a 1-factor. Since is just the union of the cycles of for , we have a decomposition of .
Theorem 1 (see ). Circulant graph for a set is connected if and only if .
In the circulant graph , is a cycle of period and length ; cycles and are of period and length each; by considering the edges , , , , and as double edges, they are cycles of period and of length , each. See Figure 2.
Proof. The proof is by induction on the order of .
Remark 4. Let . Then, the circulant graph for a set is -regular if and -regular otherwise.
Definition 5. The cross product or Cartesian product of two simple graphs and is the simple graph with vertex set in which two vertices and are adjacent if and only if either and or and .
Definition 6 (see [4, 13–18]). Circulant graphs and for and are said to be Adam’s isomorphic if there exists a positive integer relatively prime to with where , the reflexive modular reduction of a sequence , is the sequence obtained by reducing each modulo to yield and then replacing all resulting terms which are larger than by .
The following lemmas are useful to obtain one-to-one mappings.
Lemma 7. Let and be two nonempty sets. Let be a mapping. Then, is one-to-one if and only if is one-to-one for every nonempty subset of .
Lemma 8. Let and be nonempty sets and let be a partition of (each being nonempty), . Let be a mapping. Then is one-to-one if and only if is one-to-one for every .
The motivation to do this research work is to develop a theory among circulant graphs similar to the product and factorization of natural numbers.
2. Main Results
Throughout this section, the following notation is used. Let and . Then . Let denote , , and let denote disjoint union of copies of the graph . denotes the Cartesian product of graphs and .
In this paper, Cartesian product and factorization of connected circulant graphs is studied. Moreover and for and nontrivial graphs and , and in particular, for any .
In any circulant graph the length of a periodic cycle of period is . And 2, 3, 4, and 6 are the only numbers such that in any circulant graph periodic cycle of length 2, 3, 4, or 6, if it exists, occurs without rotation always. This follows from the fact that for any natural number greater than 6, there exists at least one natural number such that and so that and rotation of the periodic cycle of period is . When , is the required value. Thus, we call the periodic cycles of length 2, 3, 4, and 6 as the irrotational periodic cycles.
It is easy to verify the following:(i) for graphs , and . Here union is meant to be disjoint union only;(ii);(iii) for sets , , ;(iv)in any circulant graph removal or addition of one or more jump sizes, if possible, will not change the property of being circulant;(v)if and are nontrivial graphs of order and , respectively, , then contains number of disjoint copies of and number of disjoint copies of .
Theorem 9. Let . Then is circulant if and only if is odd. Furthermore, in that case, .
Proof. For , is a 3-regular connected graph of order . When is even, which is a disconnected graph, whereas is a connected one and hence when is even. For , contains two disjoint copies of , say and , so that and disjoint copies of . Let and , , and . Thus and and .
Let be a circulant graph with . In , cycle of period is of length 2 and irrotational. Using Remark 4, for , 3-regular connected graph is a circulant of the form for some if and only if each edge acts as a cycle of period (double edge) and equal number of vertices of (and of ) are on each sides of , . This is possible only when is odd, in which case the circulant graph is nothing but . The following transformation gives the required circulant graph representation of when is odd.
Let , . Let where the vertices are considered as the corners of a regular -gon, assumed to be located in the plane proceeding cyclically clockwise. Define the mapping such that and for every , using subscript arithmetic modulo , and . Under this mapping, , which is a periodic cycle of period and of length in , , and , using subscript arithmetic modulo , and . This implies that the mapping is a one-to-one mapping and preserves adjacency and the transformed graph is , and . Hence the result.
Note 1. Consider which is a connected circulant graph. When is even and greater than , , whereas is connected but not circulant and hence when is even and greater than 2.
Note 2. For and is not circulant since it is not a regular graph.
Note 3. For and connected circulant , since for any connected graphs and , .
Theorem 10. For , , where .
Proof. contains two disjoint copies of , say and , and disjoint copies of so that and . Let and , and so that and and . At first let us assume that is connected, . Then, for every , the length of each periodic cycle of period in is , an odd number. Thus using the proof similar to that of Theorem 9, equal number of vertices of each periodic cycle, starting from or of copies of , can be made on each sides of in , using the one-to-one mapping defined by and for every , using subscript arithmetic modulo , , and where and the vertices are considered as the corners of a regular -gon assumed to be located in the plane proceeding cyclically clockwise. Under this transformation , and , using subscript arithmetic modulo , and . And the transformed graph is , . The above result is true also when is disconnected, . Hence . When , is an odd number, hence under arithmetic modulo . Also and are Adam’s isomorphic graphs. Hence the result.
Corollary 11. For nonempty subset of , if either for any relatively prime to under arithmetic modulo or is disconnected, then connected circulant .
Proof. When and for any relatively prime to , then , using Theorem 10. When is disconnected, then the product graph is also disconnected whereas it is given that is connected. Hence the result.
Theorem 12. For and , .
Proof. At first let us assume that is connected. Let , , , and be the four copies of in so that . Let , be the copy of in and and , and .
Let and the vertices are considered as the corners of a regular -gon assumed to be located in the plane proceeding cyclically clockwise. Define one-to-one mapping such that using subscript arithmetic modulo and for every , and . Then , say, and which is a periodic cycle of period and of length in , using subscript arithmetic modulo , and . Also, is a periodic cycle of period and of length in since implies that , , and . Moreover, each cycle of period of becomes periodic cycle of period in the transformed graph. Thus the transformed graph is nothing but where . Also when , and are Adam’s isomorphic. Hence the result.
Theorem 13. Let be connected, . Then for any .
Proof. Using Theorem 9, is not circulant. For , if is connected, then is also connected and it contains disjoint copies of and number of disjoint copies of . And hence if is connected and for some , then for some and such that , and . This implies which is disconnected, a contradiction when is connected. Hence the result.
Lemma 14. Let , , and . If is not circulant, then is not circulant.
Proof. Let and . Now, for any implies that for any (if not, let and for at least one . Then by removing all other jump sizes other than from , the resultant graph, , remains circulant, a contradiction to the given condition). Similarly we can prove that for if is not circulant, then is not circulant. Combining the above arguments, we get the result.
Theorem 15. Let be connected, . Then is circulant if and only if for some . Furthermore, in that case, , where .
Proof. Let . Then , using Theorem 10. This implies that , where , using Theorem 12.
Conversely, let be circulant. If for any connected , then has as a factor and not . Let where and since is connected. Let , . Then using Theorem 2 the length of the periodic cycle of period in is , an even number. Therefore cannot be a circulant graph (see the proof of Theorem 9) and using Lemma 14, cannot be a circulant graph, a contradiction to is circulant. This implies that there exists a connected circulant such that . And thereby using Theorem 12 where . Hence the result.
Corollary 16. Let and be connected. Then is circulant if and only if is odd. Furthermore, in that case, where .
Proof. When is odd , using Theorem 12 where . Conversely, let be circulant, say for some . When is odd, the result is true by Theorem 12. When is even, using Theorem 13, is not circulant. Hence the result.
Corollary 17. Let and be connected. Then is circulant if and only if is odd if and only if is connected. Furthermore, in that case, and , where .
Theorem 18. Let and be connected. Then is circulant if and only if is odd or for some connected .
Proof. When is odd, , using Theorem 10 where . When for some connected , , using Theorem 12 where .
Conversely, let be circulant. Consider the following two cases of .
Case i. for some connected .
is connected and is circulant implies that is a connected circulant. This implies that is odd, using Corollary 16. Let so that for some connected , .
Case ii. for any connected .
Our aim is to prove that is odd in this case. If not, let , . Then is a connected circulant graph such that for any connected . This implies that , the length of a periodic cycle of period in , is odd for all since each copy of in the connected circulant graph acts as a periodic cycle of length (see the proof of Theorem 9). This is possible only when is even for every . This implies that and thereby which is not connected, a contradiction. This implies that must be odd. Hence the result.
Corollary 19. For , the products and are circulant if and only if is odd. Furthermore, in that case, and , .
Proof. For , is circulant if and only if is odd, follows from Theorem 9. Using Theorem 10, + which is a proper subgraph of , . Now consider the case of when is even. Let , . For , using Theorem 9, is not a circulant graph and hence using Lemma 14, is not a circulant graph. Hence the result.
Corollary 20. For , and for any connected circulant .
Proof. Clearly, for , , a connected circulant graph. If connected circulant graph , then, using Theorem 18, is odd or for some connected circulant , . And in both cases, using Theorems 10 and 12, is a proper subgraph of and , a contradiction for . This implies that for any connected circulant , . Similarly if connected circulant graph , then, using Corollary 16, is odd and which is a proper subgraph of and , a contradiction where and , . Hence the result.
So far we could find out when the cross product of or with another circulant is also circulant. It is interesting to know, in general, whether the cross product of any two connected circulant graphs is circulant or not. If so, when is it circulant? The following give some positive results in this direction.
Theorem 21. For , if and only if .
Proof. Let . At first, assume that . is a connected graph which implies that is also connected. This implies, using Theorem 1, which implies .
Conversely, let . Without loss of generality, let us assume that . Now is a connected 4-regular graph containing disjoint copies of , ( disjoint copies of ) and through all the isomorphic images of each vertex of , there exists a cycle of length . And for all possible values of and , and are the two edge disjoint spanning circulant subgraphs of .
contains copies of and is a spanning subgraph of . Let be the copy of in where is the isomorphic image of of in so that is a cycle of length in and , and , and . Let . Vertices are considered as the ordered vertices of a regular -gon, assumed to be located in the plane, proceeding cyclically clockwise. Define a mapping such that and for every , using subscript arithmetic modulo , and . Under this transformation which is a periodic cycle of (period ) length in , and these cycles are vertex disjoint periodic cycles in . Similarly, which is a periodic cycle of period and length in , , and these cycles are vertex disjoint periodic cycles in (transformation is similar to defined in [13–18] to define type-2 isomorphism of circulant graphs). And the edges of are the edges of disjoint copies of and disjoint copies of only. Clearly, is a one-to-one mapping and preserves adjacency and . Here the mapping is one-to-one that follows from Lemma 8 once by considering each and and in another by considering each and , and . The transformation defined on is illustrated in Figures 3 and 4. is given in Figure 3 and is given in Figure 4.
Corollary 22. Let . Then is circulant if and only if if and only if , where .
Note 4. For any integer relatively prime to and for a set , and are Adam’s isomorphic. Thus when , , . Thus , , and . Also and .
Corollary 23. Let , and . Then is not circulant.
Proof. Let and , . This implies , and which implies that ≅ and similarly for some where and . This implies that , which is not circulant using Theorem 21 since , . Hence the result.
Corollary 24. Let and be connected, , , and . Then and are circulants when and for some and .
Corollary 25. For , , , , and connected circulants , , , and , and are not circulant.
In the proof of Theorem 21, we have, when and , where . In the transformation defined on the graph each copy of the cycle is rotated uniformly (and thereby the relative positions of the vertices of each copy of are not changed) with respect to the regular -gon so as to make uniform rotation of each copy of with respect to the regular -gon. Here any vertex using subscript arithmetic modulo , and . Thereby, for , and connected circulant and , and where . And if we introduce jump sizes instead of jump size in and instead of in and making the above transformation on , we get , where and are connected, , , , , and . Thus, we obtain the following result.
Theorem 26. Let and be connected, and . Then .
Theorem 27. Let and and be connected circulant graphs. Then is circulant if and only if is circulant if and only if is circulant if and only . Furthermore, in that case, there exists natural number relatively prime to such that .
Theorem 28. Let be a connected graph of order , . Then is circulant if and only if or where is a connected circulant graph of odd order.
Proof. Let or for some connected circulant graph of odd order. Then, by Theorem 18, is circulant.
Conversely, let be a connected circulant of order for some , . Clearly, , . Now, consider different cases of .
Case i. , .
Clearly, and, using Corollary 19, . This implies that , where