ISRN Combinatorics

Volume 2013 (2013), Article ID 236298, 8 pages

http://dx.doi.org/10.1155/2013/236298

Research Article

Packings and Coverings of Various Complete Graphs with the 4-Cycle with a Pendant Edge

Brandon Coker,¹ Gary D. Coker,² Robert Gardner,³ and Yan Xia³

¹Department of Computer Science, Francis Marion University, Florence, SC 29502, USA

²Institute of Mathematics and Physics, Aberystwyth University, Aberystwyth, Ceredigion SY23 3BZ, UK

³Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614, USA

Received 17 June 2013; Accepted 12 August 2013

Academic Editors: C. da Fonseca, P. E. Jorgensen, A. V. Kelarev, and S. Zhou

Copyright © 2013 Brandon Coker et al. 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.

Abstract.

We consider the packings and coverings of complete graphs with isomorphic copies of the 4-cycle with a pendant edge. Necessary and sufficient conditions are given for such structures for (1) complete graphs , (2) complete bipartite graphs , and (3) complete graphs with a hole . In the last two cases, we address both restricted and unrestricted coverings.

1. Introduction, Motivation, and History

A -decomposition of graph is a set of subgraphs of , , where for , for , and . The are called blocks of the decomposition. The concept of a graph decomposition lies in the general area of the design theory. We can relate a graph decomposition to an experimental design by considering the following hypothetical situation: “suppose you have a collection of samples and you wish to compare a property of the samples. However, the only way to compare the samples is to run them three at a time in a machine which performs the comparison. The machine cannot be calibrated from run to run and so to compare two samples, we must run them together in the machine. When can all of the samples be optimally compared to each other by running the machine times?” The solution to this question is equivalent to finding a -decomposition of , where each vertex of represents a sample, an edge joining two vertices represents a comparison of the two corresponding samples, and a copy of represents a run of the machine. A -decomposition of exists if and only if or 3 (mod 6), and such a structure is called a Steiner triple system [1].

In the event that a -decomposition of does not exist, we can still consider a set of isomorphic copies of graphs which “approximate” a decomposition. There are two approaches to this. We describe the two approaches in terms of the sample comparison analogy. In the first approach, we can try comparing as many of the samples as possible, without repetition of comparisons (it might be that running the machine is expensive). In the setting mentioned above, we could seek a collection of runs of the machine (represented by copies of ) which do not repeat pairs of samples run together (i.e., the copies of are edge disjoint), and which minimizes the number of pairs of samples which are omitted (i.e., the cardinality of the set of edges in which are in none of the copies of is made minimal). Such an experimental design is related to a maximal graph packing. A maximal -packing of a graph with isomorphic copies of a graph is a set , where and for all , for , , and is minimal. In particular, the machine analogy corresponds to a -packing of . Such designs are explored in [2]. Other packings of the complete graphs have also been studied, for example, 4-cycle-packings [3], -packings [4], and 6-cycle-packings [5, 6]. A second approach involves comparing all of the samples to each other, but with minimal repetitions of the compared samples (we might postulate that the machine must have three samples in it during each run to keep it balanced). This experimental design is related to a minimal graph covering. A minimal -covering of a graph with isomorphic copies of a graph is a set , where , , for all , , and is minimal (when considering coverings, the graph may not be simple and may be a multiset). The machine analogy in this case corresponds to a -covering of . Such designs are explored in [7]. Coverings of have also been explored, for example, for 4-cycles [2] and 6-cycles [8].

In terms of graph decompositions, several studies have concentrated on the -decompositions of complete graphs into copies of a given graph with a small number of vertices [9–12]. In this paper, we go in a different direction and consider a single graph , the 4-cycle with a pendant edge, and explore packings and coverings of several graphs related to the complete graph. We denote the 4-cycle with a pendant edge as , where and . See Figure 1. An -decomposition of exists if and only if or 1 (mod 5), [9]. An -decomposition of the complete bipartite graph, , exists if and only if (mod 5), , and [13]. Another graph related to the complete graph is the complete graph with a hole . The complete graph on vertices with a hole of size is the graph with a vertex set , where and , and edge set . Necessary and sufficient conditions for the decomposition of into -cycles are known for [14–16]. There is an -decomposition of if and only if (mod 5), , and [13].

Figure 1: We denote by this graph.

The graph relates to the experimental design story as follows. Suppose you have performed comparisons on a collection of samples and then received an additional collection of samples (say, new samples). You now wish to compare the new samples to each other and to the original samples. In the case of the machine described above, this would correspond to a decomposition of . In the event that a decomposition does not exist, we can explore the packings and coverings of . With a maximal -packing of , we require that each copy of is a subgraph of . The definition given above for a maximal -covering also involves the condition that each copy of is a subgraph of . Most studies of coverings have involved , so the condition that the copies of are subgraphs of is trivially satisfied. But when is not a complete graph, there is no obvious reason to impose the subgraph condition. Returning to the testing-of-samples story, we see no reason to disallow, for example, the testing (or retesting) of two samples in the hole of . Therefore, we are motivated to refine the definition of a graph covering into two cases—one case in which the edges that are not in are forbidden from use in the copies of and a second case in which these edges are not forbidden. A minimal unrestricted -covering of a graph with isomorphic copies of a graph is a set where , , , and is minimal (the graph may not be simple and may be a multiset). A minimal restricted -covering of a graph with isomorphic copies of a graph is a set , where , , for all , , and is minimal. The distinction between restricted and unrestricted coverings was introduced in [17]. Notice that in the event that is a complete graph, there is no distinction between a minimal restricted and minimal unrestricted covering.

The purpose of this paper is to give -packings of , , and , as well as -coverings of , and restricted and unrestricted -coverings of and .

2. Packing and Covering

In this section, when necessary, we assume that the vertex set of is . Since has 5 vertices, we only consider .

Theorem 1. A maximal -packing of , , has leave , where , except when in which case .

Proof. Since , then it is necessary that in any -packing of with leave , (mod 5). Therefore, such a packing with (mod 5) would be maximal. If , then (mod 5), but there is not an -decomposition of [9]. So for , an -packing of with leave , where would be maximal.

Case 1. Suppose . The set is a maximal packing of with leave , where , so .

Case 2. Suppose . The set is a maximal packing of with leave , where , so .

Case 3. Suppose or 4 (mod 5), . Since (mod 5), would be optimal. Now can be decomposed [13], so .

Case 4. Suppose (mod 5), . Since (mod 5), would be optimal. Now can be decomposed [13], so .

In the following result (and throughout this paper), we refer to an equality of the form (mod ). By this, we mean that and (mod ).

Theorem 2. A minimal -covering of , , has padding , where , except when in which case .

Proof. Since , then it is necessary that in any -covering of with padding , we have (mod 5) or that (mod 5). So if (mod 5), then the covering is minimal. If , then (mod 5), but there is no -decomposition of [9]. So for , an -covering of with padding , where would be minimal.

Case 1. Suppose . The set is a minimal covering of with padding , where , so .

Case 2. Suppose . The set is a minimal covering of with padding , where , so .

Case 3. Suppose or 4 (mod 5), . There is an -decomposition of [13]. Take such a decomposition, along with another copy of which includes the edge of the hole of . This gives a covering of with padding , where (mod 5).

Case 4. Suppose (mod 5), . An -covering of is given by , with padding , where , and the covering is optimal. For , , can be decomposed [13], and can be covered with padding , where . Therefore, there is an optimal -covering of with padding , where (mod 5).

3. Packing and Covering the Complete Bipartite Graph

In this section, we consider the -packings and -coverings of the complete bipartite graph . We assume the partite sets of are and .

Theorem 3. A maximal -packing of has leave , where

(1)   if or , or if , or (2) , otherwise.

Proof. First, if or equals 1, then is not a subgraph of , and the leave must have edges. Similarly, the leave of a packing of has edges. For and , as in the proof of Theorem 1, an -packing of with leave , where (mod 5) would be maximal. Next, for and we observe that if there is a packing of with leave , then there is a packing of with leave for all . This is because , where the partite sets of are and , the partite sets of are and , the partite sets of are and , the partite sets of are and , and the partite sets of are and . There is an -decomposition of , , and [13].

In Table 1, the packings, combined with the decompositions of complete bipartite graphs mentioned above, yield the result.

Table 1

(mod 5)

Packing
Leave

,

Theorem 4. A minimal restricted -covering of , where neither nor equals and , has padding , where .

Proof. For , is not a subgraph and so a restricted -covering does not exist. Similar to the argument in Theorem 2, a -covering of with padding where (mod 5) would be minimal. As in Theorem 3, for and , if there is a restricted covering of with padding , then there is a restricted covering of with padding for all .

In Table 2, the coverings, combined with the decompositions of complete graphs mentioned in Theorem 3, yield the result.

Table 2

(mod 5)

Covering
Padding

Theorem 5. A minimal unrestricted -covering of has padding where

(1)when and , , (2)when , for , for , for .

Proof. For and , the necessary condition follows as in the proof of Theorem 4. In this case, sufficiency also follows from Theorem 4.

When , a copy of where has at most 3 edges in and at least 2 edges in the padding. So in an -covering of , there are at least copies of . Now copies of can have at most edges in and at least edges in the padding. If (mod 3), then to completely cover we must add one more copy of which has at most 1 edge in and at least 4 edges in the padding. If (mod 3), then to completely cover we must add one more copy of which has at most 2 edges in and at least 3 edges in the padding. This yields the necessary conditions for . We now establish sufficiency for .

Case 1. Suppose and (mod 3); . Consider the blocks . This is a covering of with padding , where .

Case 2. Suppose and (mod 3); . From Case 1, there is a covering of , where the partite sets of are , and with padding , where . This covering along with , is an unrestricted covering of with padding and so .

Case  3. Suppose and (mod 3); . From Case 1, there is a covering of , where the partite sets of are and with padding , where . This covering along with is an unrestricted covering of with padding , and so .

4. Packing the Complete Graph with a Hole

In this section, we assume the vertex set of is as described in Section 1, where and .

Theorem 6. A maximal -packing of has leave , where and is necessary.

Proof. When , is not a subgraph of , and so, there is no packing. Therefore, is necessary for the existence of a packing.

Case  1. If , then , where the vertex set of is and the partite sets of are and . There exists a packing with leave such that . Without loss of generality, . Take such a packing along with . This yields a packing of with leave , so .

Case 2. Suppose (mod 5) and (mod 5). Then , where and the partite sets of are and . We have (mod 5) and (mod 5). There is a maximal packing of , where (mod 5) with by Theorem 1 and a maximal packing of with by Theorem 3. Therefore, there is a maximal packing of with leave , where (mod .

Case 3. Suppose (mod 5) and (mod 5) or (mod 5) and (mod 5). Then as in Case 2, where (mod 5) and (mod 5) or (mod 5) and (mod 5). There is a maximal packing of , where (mod 5) or (mod 5) with by Theorem 1, and there is a maximal packing of with by Theorem 3. Therefore, there is a maximal packing of with leave , where (mod 5).

Case 4. Suppose (mod 5) and (mod 5). When , , as in Case 2, where (mod 5) and (mod 5). There is a maximal packing of with by Theorem 3 and , where (mod 5) is decomposable [9]. Therefore, there is a maximal packing of with leave , where (mod 5).

Case 5. Suppose (mod 5) and (mod 5) or (mod 5) and (mod 5). Then , as in Case 2, where (mod 5) and (mod 5), or (mod 5) and (mod 5). There is a maximal packing of , where (mod 5) or (mod 5) with by Theorem 1, and there is a maximal packing of with by Theorem 3. Therefore, there is a maximal packing of with leave , where .

Case 6. Suppose (mod 5) and (mod 5). Then , where and the partite sets of are and . Then there is a maximal packing of with leave , where by Theorem 1, and there is a maximal packing of with leave , where by Theorem 3. Therefore, there is a maximal packing of with leave , where (mod 5).

Case 7. Suppose (mod 5) and (mod 5) or (mod 5), and (mod 5). Then where (mod 5) and (mod 5) or (mod 5), and (mod 5). There is a maximal packing of , where (mod 5) or (mod 5) with by Theorem 1 and is decomposable [13]. Therefore, there is a maximal packing of with leave where (mod 5).

Case 8. Suppose (mod 5) and (mod 5). Similar to Case 3, when , , as in Case 2, where (mod 5) and (mod 5). There is a maximal packing of with by Theorem 3 and , where (mod 5) is decomposable [9]. Therefore, there is a maximal packing of with leave , where (mod 5).

Case 9. Suppose (mod 5) and (mod 5), or (mod 5) and (mod 5). Then , as in Case 2, where (mod 5), (mod 5) or (mod 5) and (mod 5). There is a maximal packing of , where (mod 5) or (mod 5) with by Theorem 1, and there is a maximal packing of with by Theorem 3. Therefore, there is a maximal packing of with leave , where (mod 5).

Case 10. Suppose (mod 5) and (mod 5). Then , as in Case 2, where (mod 5) and (mod 5). There is a maximal packing of , where (mod 5) with by Theorem 1 and is decomposable [13]. Therefore, there is a maximal packing of with leave , where (mod 5).

Case 11. Suppose (mod 5) and (mod 5). Similar to Case 4, when , , where (mod 5) and (mod 5). There is a maximal packing of with by Theorem 3 and where (mod 5) is decomposable [9]. Therefore, there is a maximal packing of with leave , where (mod 5).

Case 12. Suppose (mod 5) and (mod 5). As in Case 6, we have . Then there is a maximal packing of with leave , where by Theorem 1 and is decomposable [13]. There is a maximal packing of with leave , where (mod 5).

Case 13. Suppose (mod 5) and (mod 5). Similar to Case 4, when , consider , where (mod 5) and (mod 5). There is a maximal packing of with by Theorem 3 and , where (mod 5) is decomposable [9]. Therefore, there is a maximal packing of with leave , where (mod .

5. Covering the Complete Graph with a Hole

As in the previous section, we assume the vertex set of is , where and .

Theorem 7. A minimal restricted -covering of has padding , where , when . No restricted -covering of exists for .

Proof. First, suppose . Consider the edge . If is the pendant edge of an , say , then , and must be distinct vertices in . But , so this cannot happen. If is an edge in the 4-cycle of some , then there must be an edge in the 4-cycle of the form , a contradiction to the restricted covering. So, is necessary.

Similar to the argument in Theorem 2, an -covering of with padding where (mod 5) would be minimal.

Case 1. Suppose (mod 5) and (mod 5). First, can be covered with , and this has a padding with and so . For general and , , where the vertex set of is and the hole is on vertex set , the partite sets of are and , and the partite sets of are and . Now, and can be decomposed [13]. Taking these decompositions along with the above covering of yields a covering of with padding , where , and so (mod 5).

Case 2. Suppose (mod 5) and (mod 5) or (mod 5), and (mod 5). Consider , where and the partite sets of are and and (mod 5) and (mod 5) or (mod 5), and (mod 5). There is a maximal packing of where (mod 5) or (mod 5) with by Theorem 1. There is a maximal packing of with by Theorem 3. Therefore, there is a minimal covering of with padding , where (mod 5).

Case 3. Suppose (mod 5), and (mod 5). Consider , as in Case 2, where (mod 5), and (mod 5). There is a minimal covering of with padding , where by Theorem 4 and , where (mod 5) is decomposable [9]. Therefore, there is a minimal covering of with padding , where (mod 5).

Case  4. Suppose (mod 5) and (mod 5). Consider , as in Case 2, where (mod 5) and (mod 5). There is a maximal packing of with leave , where by Theorem 1 and, without loss of generality, . There is a maximal packing of with leave , where and by Theorem 3. These two packings combined with yield a covering of with padding , where , so (mod 5).

Case 5. Suppose (mod 5) and (mod 5). Consider , as in Case 2, where (mod 5) and (mod 5). There is a maximal packing of with leave , where by Theorem 1 and, without loss of generality, = . There is a maximal packing of with leave , where and by Theorem 3. These two packings combined with yield a covering of with padding , where , so (mod 5).

Case 6. Suppose (mod 5) and (mod 5). Consider , where , and the partite sets of are and . Then there is a maximal packing of with leave , where by Theorem 1, and there is a maximal packing of with leave , where by Theorem 3. Therefore, we can add an additional copy of which includes the edges in and . So, there is a minimal covering of with padding , where (mod 5).

Case 7. Suppose (mod 5) and (mod 5), or (mod 5) and (mod 5). Consider , as in Case 2, where (mod 5), and (mod 5) or (mod 5), and (mod 5). There is a minimal covering of with padding , where by Theorem 2, and is decomposable [13]. Therefore, there is a minimal covering of with padding where (mod 5).

Case 8. Suppose (mod 5) and (mod 5). Consider , as in Case 2, where (mod 5) and (mod 5). There is a minimal covering of with padding , where by Theorem 4 and , where (mod 5) is decomposable [9]. Therefore, there is a minimal covering of with padding , where (mod 5).

Case 9. Suppose (mod 5) and (mod 5). Consider , as in Case 2, where (mod 5) and (mod 5). There is a maximal packing of with leave , where by Theorem 1 and, without loss of generality, = . There is a maximal packing of with leave , where and by Theorem 3. These two packings combined with yield a covering of with padding , where , so (mod 5).

Case 10. Suppose (mod 5) and (mod 5). Consider , as in Case 2, where (mod 5) and (mod 5). There is a maximal packing of with leave , where by Theorem 1 and, without loss of generality, = . These is a maximal packing of with leave , where and by Theorem 3. There two packings combined with yield a covering of with padding , where , so (mod 5).

Case 11. Suppose (mod 5) and (mod 5). Consider , as in Case 2, where (mod 5) and (mod 5). There is a minimal covering of with padding , where by Theorem 2, and is decomposable [13]. Therefore, there is a minimal covering of with padding , where (mod 5).

Case 12. Suppose (mod 5) and (mod 5). Consider , as in Case 2, where (mod 5) and (mod 5). There is a minimal covering of with padding , where by Theorem 4 and , where (mod 5) is decomposable [9]. Therefore, there is a minimal covering of with padding , where (mod 5).

Case 13. Suppose (mod 5) and (mod 5). As in Case 6, we have . Then there is a minimal covering of with padding , where by Theorem 2 and is decomposable [13]. Therefore, there is a minimal covering of with padding , where (mod 5).

Case 14. Suppose (mod 5) and (mod 5). Consider , as in Case 2, where (mod 5) and (mod 5). There is a minimal covering of with padding , where by Theorem 4 and , where (mod 5) is decomposable [9]. Therefore, there is a minimal covering of with padding , where (mod 5).

Theorem 8. A minimal unrestricted -covering of has padding where

(1)when , , (2)when , for , for , for , (3)when , , where for and for .

Proof. When , the necessary and sufficient conditions follow from Theorem 7. When , and the necessary and sufficient conditions follow from Theorem 5.

When , similar to the argument in Theorem 2, an -covering of with padding must satisfy (mod 5). Since an -decomposition of does not exist for (mod 5) [13], the necessary conditions follow for and . For , since , then an unrestricted -covering of with padding where would be minimal. However, in such a covering, there are only two copies of . Edge cannot be the pendant edge of a copy of in such a covering since this copy would have 2 edges in the padding. If edge is in a copy of and is not the pendant edge, then this copy of must be of the form for some distinct . However, the complement of this graph in is not a copy of . Therefore, no such -covering of exists, and a minimal unrestricted -covering of with padding , where would be minimal. The set is an unrestricted -covering of with padding where . So , and the covering is minimal.

Case  1. Suppose and (mod 5); . Then where the vertex set of is and the hole is on vertex set , and the partite sets of are and . There is an -decomposition of [13], and the set is an unrestricted -covering of with padding , where , and . So, there is an unrestricted covering of with padding , where (mod 5).

Case  2. Suppose , (mod 5); . Then, as in Case 1, . There is an -decomposition of [13], and the set is an unrestricted -covering of with padding , where and . So, there is an unrestricted covering of with padding , where (mod 5).

Case 3. Suppose and (mod 5); . Then, as in Case 1, . There is an -decomposition of [13], and the set , is an unrestricted -covering of with padding , where and . So, there is an unrestricted covering of with padding , where , where (mod 5).

Case 4. Suppose and (mod 5). Then, as in Case 1, . There is an -decomposition of [13], and the set is an unrestricted -covering of with padding , where and . So, there is an unrestricted covering of with padding where (mod 5).

Case 5. Suppose and (mod 5); . Then, as in Case 1, . There is an -decomposition of [13], and the set , , is an unrestricted covering of with padding , where and . So, there is an unrestricted covering of with padding , where (mod 5).

6. Conclusion

Motivated by experimental designs and comparisons of samples, we have given necessary and sufficient conditions for the -packings and -coverings of complete graphs, complete bipartite graphs, and complete graphs with a hole, where is a 4-cycle with a pendant edge. For complete bipartite graphs and complete graphs with a hole, we have given both restricted and unrestricted coverings.

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