Abstract
Let be a graph and let be a subgraph of . Assume that has an -decomposition such that for all . An -supermagic decomposition of is a bijection such that is a constant for each in the decomposition and . If admits an -supermagic decomposition, then is called -supermagic decomposable. In this paper, we give necessary and sufficient conditions for the existence of -supermagic decomposition of the complete bipartite graph minus a one-factor.
1. Introduction
All graphs in this paper are finite and simple. We use terminologies of graphs from West’s textbook [1]. If and are integers with , we denote by .
Let be a graph. A covering of is a family of subgraphs such that each edge of belongs to at least one of the subgraphs . If , for all , then the covering is called an -covering of .
Assume that has an -covering. An -magic labeling of is a bijection such that is a constant for each in the -covering, and is called a magic constant. An -magic labeling is called an -supermagic labeling if . A graph is said to be -supermagic if admits an -magic labeling.
The -supermagic labeling was first introduced by Gutiérrez and Lladó [2] in 2005. They proved that many classes of graphs are -supermagic, such as the complete bipartite graphs and the star are -supermagic for some positive integer . In 2007, Lladó and Moragas [3] studied some windmills, wheels, and thetas which are -supermagic. For example, they showed that the wheel for odd is -supermagic.
In 2010, Ngurah et al. [4] proved that the fan , the triangle ladder , and are -supermagic. Furthermore, they showed that the book , the ladder , and the grid are -supermagic.
An -decomposition of a graph is an -covering of which forms a partition of the edge set of . If has an -decomposition, then is called -decomposable. Assume that is -decomposable. If admits an -supermagic labeling, then is said to be -supermagic decomposable or has an -supermagic decomposition.
The concept of an -supermagic decomposition of arises from the combination between graph labelings and graph decompositions which was introduced by Liang [5] in 2012. He found the conditions for the existence of -supermagic decomposition of the complete -partite graph as well as of multiple copies of it. Moreover, in 2015, Marimuthu and Kumar [6] showed that the complete bipartite graph is -supermagic decomposable. For more information about graph labelings, please see Gallian’s survey [7].
Motivated by the results of these authors, we are going to find necessary and sufficient conditions for the existence of an -supermagic decomposition of the complete bipartite graph minus a one-factor, where . Hereafter, if , for some positive integer , then is called an -star subgraph. The notation of an -star-subgraph was introduced by Akiyama and Kano [8].
2. Main Result
In this section, we consider the graph , where is a one-factor of and . Note that is -star-decomposable for all . Since and , we have .
Theorem 1. Let be a positive integer with . Then is not an -star-supermagic decomposable graph.
Proof. Suppose that is -star-supermagic decomposable. Let be an -star-supermagic labeling of with the magic constant . Let and be vertex-bipartitions of . Let be an -star-decomposition of , where contains one vertex of and vertices of for all .
Consider the labeling ; assume that all vertices of are labeled by and all vertices of are labeled by . Note that for all . Then Since , we have but ; that is, for some integer . Note that is odd and is even. Thus, is odd. We know that because the magic constant is . Since is even, this is impossible and results in a contradiction to our assumption. Hence, is not an -star-supermagic decomposable graph.
The necessary condition for the existence of an -star-supermagic decomposition of is settled directly from Theorem 1.
Corollary 2. If is -star-supermagic decomposable, then is odd or .
Next we establish a structural lemma which will be used in the proof of Theorem 4.
Lemma 3. For any positive integer , the set can be partitioned into elements subsets which satisfy two conditions as follows:(i)There are subsets of such that the summation of all elements in each subset is .(ii)There are subsets of such that the summation of all elements in each subset is .
Proof. For , let , let , and let .
Define by Let for all . Then the set Now, let , let , and let and let , let , and let . Thus, .
Let for all . Then .
Define by for all . For each , there exists an element such that , so . Then we have Hence, there are subsets of such that the summation of all elements in each subset is , and there are subsets of such that the summation of all elements in each subset is .
Theorem 4. Let be a positive integer with . Then is an -star-supermagic decomposable graph with the magic constant .
Proof. Let for some positive integer , and let and be vertex-bipartition of . Let be an -star-decomposition of , where with and for all .
Define a total labeling byfor , and for , where .
Note thatNow, edges of each have been labeled with the partial magic sum , and their edges are labeled with values from . There are still three edges within each which need to be labeled, and their label will come from . By applying Lemma 3 to the labeling , we have the following properties: (i)There are subsets of such that the summation of all elements in each subset is .(ii)There are subsets of such that the summation of all elements in each subset is .Thus, has the summation of its three remaining edges as for and for .
Hence, for ,and, for ,Therefore is an -star-supermagic decomposable graph, where with the magic constant .
Example 5. Consider the graph . Let and be vertex-bipartition of . We have that is -star-supermagic decomposable by using the labeling in Table 1 with the magic constant .
Theorem 6. Let be a positive integer, where is odd and . Then is an -star-supermagic decomposable graph with the magic constant .
Proof. Let and be vertex-bipartition of . Let be an -star-decomposition of , where with and for all .
Define a total labeling by for , and for , where .
To get the magic constant from the labeling, we compute for all . Hence is an -star-supermagic decomposable graph for all odd .
The following corollary is a direct consequence of Corollary 2 and Theorems 4–6.
Corollary 7. Let be a positive integer, where . Then has an -star-supermagic decomposition if and only if is odd or .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-60-GEN-012).