A graph G with q edges is said to be harmonious, if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x)+f(y) (mod q), 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 G and for the corona graph of K2 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 (p,q)-graph G of order p and size q to be harmonious, if there is an injective function f:V(G)→ℤq, where ℤq is the group of integers modulo q, such that the induced function f*:E(G)→ℤq, 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.
When G is a tree or, in general for a graph G with p=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 2k, then q is divisible by 2k+1. 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 ℤq and b is any element of ℤq.
Chang et al. [2] define an injective labeling f of a graph G 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 0 up 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 K2 and a tree or K2 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 G of order p and H, the corona of G with H, denoted by G⊙H, is the graph with V(G⊙H)=V(G)∪⋃i=1pV(Hi), and E(G⊙H)=E(G)∪⋃i=1p(E(Hi)∪{(vi,u):vi∈V(G)andu∈V(Hi)}). In other words, a corona graph is obtained from two graphs, G of order p and H, taking one copy of G and p copies of H and joining by an edge the ith vertex of G to every vertex in the ith copy of H.
Grace [4] showed that C2n+1⊙K1 is harmonious and conjectured that C2n⊙K1 is harmonious. This conjecture has been proved by Liu and Zhang [8] and Liu [9]. Singh in [10, 11] has proved that Cn⊙K2 and Cn⊙K3 are sequential for all odd n>1. Santhosh [12] has shown that Cn⊙P4 is sequential for all odd n≥3.
The join of two graphs G and H, denoted by G+H, is the graph where V(G)∩V(H)=∅ and each vertex of G is adjacent to all vertices of H. When H=K1, this is the corona graph K1⊙G.
Graham and Sloane [1] showed harmonious labeling of the join of the path Pn and K1, that is, the fan Fn=Pn+K1, and harmonious labeling of the double fanPn+K2¯. Later, Chang et al. [2] gave harmonious labeling of the join of the star Sn and K1.
The next result shows that if join of a graph G and K1 is strongly harmonious, then the corona of a cycle and the graph G admitted harmonious labeling.
Theorem 1.
Let G be a graph of order p and size q. If G+K1 is strongly harmonious with the 0 label on the vertex of K1, then Cn⊙G is harmonious for all odd n≥3.
Proof.
Let G be a (p,q)-graph and G+K1 strongly harmonious with the 0 label on the vertex x∈K1. Then, there exists labeling f:V(G+K1)→{0,…,p+q-1} such that f(x)=0 and the edge labels are from the set {f*(uv)=f(u)+f(v):uv∈E(G+K1)}={1,2,…,p+q}.
Now, for n odd, n≥3, we consider the corona graph Cn⊙G with n(p+1) vertices and Γ=n(p+q+1) edges. Denote the vertices and edges of the cycle Cn such that V(Cn)={x1,x2,…,xn} and E(Cn)={xixi+1:1≤i≤n-1}∪{xnx1}. By the symbol yi, we denote a vertex in the ith copy of G, denoted by Gi, corresponding to the vertex y in G; that is, y∈V(G) and yi∈V(Gi).
We define the vertex labeling g:V(Cn⊙G)→{0,1,…,Γ-1} in the following:
(1)g(xi)=(p+q+1)(i-1),for1≤i≤n,g(yi)=f(y)+(p+q+1)(i-1),for1≤i≤n.
If we denote the join graph G+K1 as G+{x}, then the set of all edge labels of the ith copy of G+{x} consists of the consecutive integers g*(E(Gi+{xi}))={2(p+q+1)(i-1)+1,2(p+q+1)(i-1)+2,…,2(p+q+1)(i-1)+p+q}, 1≤i≤n. For edge labels of the cycle Cn, we have g*(xixi+1)=g(xi)+g(xi+1)=(p+q+1)(2i-1), for 1≤i≤n-1, and g*(xnx1)=g(xn)+g(x1)=(p+q+1)(n-1).
It is not difficult to see that, for 1≤i≤(n-1)/2, it is true that
1+max{g*(E(G((n+1)/2)+i+{x((n+1)/2)+i}))}=2(p+q+1)i (mod Γ) and it is equal to g*(x((n+1)/2)+ix((n+1)/2)+i+1) (mod Γ);
1+g*(x((n+1)/2)+ix((n+1)/2)+i+1) (mod Γ) is equal to the min{g*(E(Gi+1+{xi+1}))}=2(p+q+1)i+1.
Moreover, 1+max{g*(E(G(n+1)/2+{x(n+1)/2}))}=0 (mod Γ) and it is equal to g*(x(n+1)/2x(n+3)/2)=0 (mod Γ).
Thus, under the induced mapping g*, all the resulting edge labels are distinct and they get the consecutive integers from 0 up to n(p+q+1)-1 (mod Γ). This concludes the proof.
Graham and Sloane [1] have proved that the fans Fm=Pm+K1, m≤7, and the wheels Wm=Cm+K1, m≢2 (mod 3), are strongly harmonious with the 0 label on the vertex of K1. In light of these results and Theorem 1, we have the following corollaries.
Corollary 2.
Let Cn⊙Pm be the corona graph of a cycle Cn and a path Pm. Then, Cn⊙Pm is harmonious for all odd n≥3 and 1≤m≤7.
Corollary 3.
Let Cn⊙Cm be the corona graph of two cycles. Then, Cn⊙Cm is harmonious for all odd n≥3 and m≢2 (mod3).
Shee [13] has shown that the complete tripartite graph K1,m,k=Km,k+K1, m,k≥1, is strongly harmonious, while Gnanajothi [14] proved that K1,1,m,k=K1,m,k+K1, m,k≥1, is also strongly harmonious. In both cases, the vertex of K1 is labeled by the 0 label. Thus, with respect to Theorem 1, we obtain the following.
Corollary 4.
For m,k≥1 and odd n≥3, the corona graph Cn⊙Km,k is harmonious.
Corollary 5.
For m,k≥1 and odd n≥3, the corona graph Cn⊙K1,m,k is harmonious.
Let one consider the graphs obtained by corona operation between the single edge K2 and a tree.
Theorem 6.
If T is a strongly c-harmonious tree of odd size q and c=(q+1)/2, then the corona graph K2⊙T is also strongly c-harmonious.
Proof.
Let T be a tree of size q with strongly c-harmonious labeling f:V(T)→{0,1,…,q}, where the edge labels are from the set of consecutive integers {f*(e):e∈E(T)}={c,c+1,…,c+q-1}.
Consider the corona graph K2⊙T with vertices x1,x2∈V(K2) and vertices yi∈V(Ti), i=1,2, corresponding to the vertices y∈T, where the vertex xi is incident to every vertex in Ti for i=1,2.
Define now new vertex labeling g:V(K2⊙T)→{0,1,…,4q+2} such that
(2)g(xi)={c+q,fori=1,q+1,fori=2,g(yi)={f(y),fori=1andeveryy∈T,f(y)+c+q+1,fori=2andeveryy∈T.
Thus, Im(g)={0,1,2,…,q,q+1}∪{c+q,c+q+1,c+q+2,…,c+2q,c+2q+1} and, for the edge labels, we have
(3){g*(e):e∈E(T1)}={c,c+1,c+2,…,c+q-1},{g*(x1y1)=g(x1)+g(y1):y1∈V(T1)}={c+q,c+q+1,…,c+2q},g*(x1x2)=g(x1)+g(x2)=c+2q+1,{g*(x2y2)=g(x2)+g(y2):y2∈V(T2)}={c+2q+2,c+2q+3,…,c+3q+2},{g*(e):e∈E(T2)}={3c+2q+2,3c+2q+3,…,3c+3q+1}.
We can see that edge labels form the set of consecutive integers from c up to 3c+3q+1 if and only if max{g*(x2y2)=g(x2)+g(y2):y2∈V(T2)}+1=min{g*(e):e∈E(T2)}; that is, c=(q+1)/2.
We know that every caterpillar Catp admits strongly c-harmonious labeling. As an illustration, Figure 1 provides an example of the strongly 5-harmonious labeling of Cat10.
Strongly 5-harmonious labeling of the caterpillar Cat10.
As an immediate consequence of Theorem 6, we can state the following corollary.
Corollary 7.
Let
Cat
q+1 be a caterpillar of odd size q. If
Cat
q+1 admits strongly (q+1)/2-harmonious labeling, then the corona graph K2⊙
Cat
q+1 also admits strongly (q+1)/2-harmonious labeling.
Theorem 8.
Let G be a unicyclic graph of odd size q. If G is a strongly c-harmonious and c=(q-1)/2, then the corona graph K2⊙G is also strongly c-harmonious.
Proof.
Let G be a connected (p,q)-graph containing exactly one cycle. Clearly, p=q. Let f:V(G)→{0,1,…,q-1} be strongly c-harmonious labeling with the edge labels from the set of consecutive integers {f*(e):e∈E(G)}={c,c+1,…,c+q-1}.
If x1 and x2 are the vertices of K2 and if by the symbol yi we mean a vertex in the ith copy of G corresponding to the vertex y∈V(G), then sets of vertices and edges of the corona graph K2⊙G are as follows: V(K2⊙G)=V(K2)∪V(G1)∪V(G2), E(K2⊙G)={x1x2}∪E(G1)∪{x1y1:y1∈V(G1)}∪E(G2)∪{x2y2:y2∈V(G2)}.
Define new vertex labeling g:V(K2⊙G)→{0,1,…,4q} in the following:
(4)g(xi)={c+q,fori=1,q,fori=2,g(yi)={f(y),fori=1andeveryy∈G,f(y)+c+q+1,fori=2andeveryy∈G.
The image of the vertex labeling g is a union of two sets of consecutive integers Im(g)={0,1,2,…,q}∪{c+q,c+q+1,c+q+2,…,c+2q}. Observe that the edge labels are
(5){g*(e):e∈E(G1)}={c,c+1,c+2,…,c+q-1},{g*(x1y1)=g(x1)+g(y1):y1∈V(G1)}={c+q,c+q+1,…,c+2q-1},g*(x1x2)=g(x1)+g(x2)=c+2q,{g*(x2y2)=g(x2)+g(y2):y2∈V(T2)}={c+2q+1,c+2q+2,…,c+3q},{g*(e):e∈E(T2)}={3c+2q+2,3c+2q+3,…,3c+3q+1}.
The edge labels form the set of consecutive integers from c up to 3c+3q+1 if and only if c+3q+1=3c+2q+2. It is true if c=(q-1)/2. Thus, the labeling g is strongly (q-1)/2-harmonious labeling of the corona graph K2⊙G.
An example of the strongly 4-harmonious unicyclic graph is presented in Figure 2.
Strongly 4-harmonious labeling of a unicyclic graph.
We know that every odd cycle C2n+1 admits strongly n-harmonious labeling. As consequence of Theorem 8, we have the following.
Corollary 9.
The corona graph K2⊙C2n+1, n≥1, is strongly n-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.
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