On Harmonious Labeling of Corona Graphs

Harmonious graphs naturally arose in the study of modular version of error-correcting codes and channel assignment problems. Graham and Sloane [1] defined a (p, q)-graph G of order p and size q to be harmonious, if there is an injective function f : V(G) → Z q , where Z q is the group of integers modulo q, such that the induced function f : E(G) → Z q , defined by f(xy) = f(x) + f(y) for each edge xy ∈ E(G), is a bijection. The function f is called harmonious labeling and the image of f denoted by Im(f) is called the corresponding set of vertex labels. WhenG is a tree or, in general for a graphGwithp = q+1, 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 q and the degree of every vertex is divisible by 2, then q is divisible by 2. This necessary condition is called the harmonious parity condition. They also proved that if f is harmonious labeling of a graph G of size q, then so is af + b labeling, where a is an invertible element of Z q and b is any element of Z q . Chang et al. [2] define an injective labeling f of a graphG with q edges to be strongly c-harmonious, if the vertex labels are from the set {0, 1, . . . , q − 1} and the edge labels are from the set {f(xy) = f(x)+f(y) : xy ∈ E(G)} = {c, c+1, . . . , c+ q − 1}. Grace [3, 4] called such labeling sequential. In the case of a tree, Grace allows the vertex labels to range from 0up to q. Strongly 1-harmonious graph is called strongly harmonious. By taking the edge labels of a sequentially labeled graph with q edges modulo q, 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 G and also between K 2 and a tree or K 2 and a unicyclic graph.


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  : () → Z  , where Z  is the group of integers modulo , such that the induced function  * : () → Z  , defined by  * () = () + () for each edge  ∈ (), is a bijection.
The function  is called harmonious labeling and the image of  denoted by Im() is called the corresponding set of vertex labels.
When  is a tree or, in general for a graph  with  = +1, 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 2  , then  is divisible by 2 +1 .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 Z  and  is any element of Z  .
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  2 and a tree or  2 and a unicyclic graph.

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 ( ⊙ ) = () ∪ ⋃  =1 (  ), and ( ⊙ ) = () ∪ ⋃  =1 ((  ) ∪ {(V  , ) : V  ∈ () 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 .
The join of two graphs  and , denoted by  + , is the graph where ()∩() = 0 and each vertex of  is adjacent to all vertices of .When  =  1 , this is the corona graph  1 ⊙ .
Graham and Sloane [1] showed harmonious labeling of the join of the path   and  1 , that is, the fan   =   + 1 , and harmonious labeling of the double fan   +  2 .Later, Chang et al. [2] gave harmonious labeling of the join of the star   and  1 .
The next result shows that if join of a graph  and  1 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  +  1 is strongly harmonious with the 0 label on the vertex of  1 , then   ⊙  is harmonious for all odd  ≥ 3.
Corollary 2. Let   ⊙  be the corona graph of a cycle   and a path   .Then,   ⊙   is harmonious for all odd  ≥ 3 and 1 ≤  ≤ 7.
Shee [13] has shown that the complete tripartite graph  1,, =  , +  1 , ,  ≥ 1, is strongly harmonious, while Gnanajothi [14] proved that  1,1,, =  1,, +  1 , ,  ≥ 1, is also strongly harmonious.In both cases, the vertex of  1 is labeled by the 0 label.Thus, with respect to Theorem 1, we obtain the following.Let one consider the graphs obtained by corona operation between the single edge  2 and a tree.Theorem 6.If  is a strongly -harmonious tree of odd size  and  = ( + 1)/2, then the corona graph  2 ⊙  is also strongly -harmonious.
If  1 and  2 are the vertices of  2 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  2 ⊙  are as follows: Define new vertex labeling  : ( 2 ⊙) → {0, 1, . . ., 4} in the following: for  = 1 and every  ∈ ,  () +  +  + 1, for  = 2 and every  ∈ .
An example of the strongly 4-harmonious unicyclic graph is presented in Figure 2.