Abstract
A graph with edges is said to be harmonious, if there is an injection from the vertices of to the group of integers modulo such that when each edge is assigned the label (mod ), the resulting edge labels are distinct. In this paper, we study the existence of harmonious labeling for the corona graphs of a cycle and a graph and for the corona graph of and a tree.
1. Introduction
Harmonious graphs naturally arose in the study of modular version of error-correcting codes and channel assignment problems. Graham and Sloane [1] defined a -graph of order and size to be harmonious, if there is an injective function , where is the group of integers modulo , such that the induced function , defined by for each edge , is a bijection.
The function is called harmonious labeling and the image of denoted by is called the corresponding set of vertex labels.
When is a tree or, in general for a graph with , exactly one label may be used on two vertices.
Graham and Sloane [1] proved that if a harmonious graph has an even number of edges and the degree of every vertex is divisible by , then is divisible by . This necessary condition is called the harmonious parity condition. They also proved that if is harmonious labeling of a graph of size , then so is labeling, where is an invertible element of and is any element of .
Chang et al. [2] define an injective labeling of a graph with edges to be strongly -harmonious, if the vertex labels are from the set and the edge labels are from the set . Grace [3, 4] called such labeling sequential. In the case of a tree, Grace allows the vertex labels to range from up to . Strongly -harmonious graph is called strongly harmonious.
By taking the edge labels of a sequentially labeled graph with edges modulo , we obviously obtain a harmoniously labeled graph. It is not known if there is a graph that can be harmoniously labeled but not sequentially labeled. More than 50 papers have been published on harmonious labeling and comprehensive information can be found in [5]. Similarly, labeling of special types of crown graphs is examined in [6].
In this paper, we study the existence of harmonious labeling for the graphs obtained by corona operation between a cycle and a graph and also between and a tree or and a unicyclic graph.
2. Main Results
In this section, we present the results related to corona graphs. The corona operation between two graphs was introduced by Frucht and Harary [7]. Given two graphs of order and , the corona of with , denoted by , is the graph with and . In other words, a corona graph is obtained from two graphs, of order and , taking one copy of and copies of and joining by an edge the th vertex of to every vertex in the th copy of .
Grace [4] showed that is harmonious and conjectured that is harmonious. This conjecture has been proved by Liu and Zhang [8] and Liu [9]. Singh in [10, 11] has proved that and are sequential for all odd . Santhosh [12] has shown that is sequential for all odd .
The join of two graphs and , denoted by , is the graph where and each vertex of is adjacent to all vertices of . When , this is the corona graph .
Graham and Sloane [1] showed harmonious labeling of the join of the path and , that is, the fan , and harmonious labeling of the double fan . Later, Chang et al. [2] gave harmonious labeling of the join of the star and .
The next result shows that if join of a graph and is strongly harmonious, then the corona of a cycle and the graph admitted harmonious labeling.
Theorem 1. Let be a graph of order and size . If is strongly harmonious with the label on the vertex of , then is harmonious for all odd .
Proof. Let be a -graph and strongly harmonious with the label on the vertex . Then, there exists labeling such that and the edge labels are from the set .
Now, for odd, , we consider the corona graph with vertices and edges. Denote the vertices and edges of the cycle such that and . By the symbol , we denote a vertex in the th copy of , denoted by , corresponding to the vertex in ; that is, and .
We define the vertex labeling in the following:
If we denote the join graph as , then the set of all edge labels of the th copy of consists of the consecutive integers , . For edge labels of the cycle , we have , for , and .
It is not difficult to see that, for , it is true that(i);(ii) (mod );(iii) (mod ) and it is equal to (mod );(iv) (mod ) is equal to the .Moreover, (mod ) and it is equal to (mod ).
Thus, under the induced mapping , all the resulting edge labels are distinct and they get the consecutive integers from up to (mod ). This concludes the proof.
Graham and Sloane [1] have proved that the fans , , and the wheels , (mod ), are strongly harmonious with the label on the vertex of . In light of these results and Theorem 1, we have the following corollaries.
Corollary 2. Let be the corona graph of a cycle and a path . Then, is harmonious for all odd and .
Corollary 3. Let be the corona graph of two cycles. Then, is harmonious for all odd and (mod).
Shee [13] has shown that the complete tripartite graph , , is strongly harmonious, while Gnanajothi [14] proved that , , is also strongly harmonious. In both cases, the vertex of is labeled by the label. Thus, with respect to Theorem 1, we obtain the following.
Corollary 4. For and odd , the corona graph is harmonious.
Corollary 5. For and odd , the corona graph is harmonious.
Let one consider the graphs obtained by corona operation between the single edge and a tree.
Theorem 6. If is a strongly -harmonious tree of odd size and , then the corona graph is also strongly -harmonious.
Proof. Let be a tree of size with strongly -harmonious labeling , where the edge labels are from the set of consecutive integers .
Consider the corona graph with vertices and vertices , , corresponding to the vertices , where the vertex is incident to every vertex in for .
Define now new vertex labeling such that
Thus, and, for the edge labels, we have
We can see that edge labels form the set of consecutive integers from up to if and only if ; that is, .
We know that every caterpillar admits strongly -harmonious labeling. As an illustration, Figure 1 provides an example of the strongly -harmonious labeling of .
As an immediate consequence of Theorem 6, we can state the following corollary.
Corollary 7. Let be a caterpillar of odd size . If admits strongly -harmonious labeling, then the corona graph also admits strongly -harmonious labeling.
Theorem 8. Let be a unicyclic graph of odd size . If is a strongly -harmonious and , then the corona graph is also strongly -harmonious.
Proof. Let be a connected -graph containing exactly one cycle. Clearly, . Let be strongly -harmonious labeling with the edge labels from the set of consecutive integers .
If and are the vertices of and if by the symbol we mean a vertex in the th copy of corresponding to the vertex , then sets of vertices and edges of the corona graph are as follows: , .
Define new vertex labeling in the following:
The image of the vertex labeling is a union of two sets of consecutive integers . Observe that the edge labels are
The edge labels form the set of consecutive integers from up to if and only if . It is true if . Thus, the labeling is strongly -harmonious labeling of the corona graph .
An example of the strongly -harmonious unicyclic graph is presented in Figure 2.
We know that every odd cycle admits strongly -harmonious labeling. As consequence of Theorem 8, we have the following.
Corollary 9. The corona graph , , is strongly -harmonious.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments and suggestions leading to improvement of this paper. The research for this paper was supported by Slovak VEGA Grant 1/0130/12.