ON k-GRACEFUL LABELING OF SOME GRAPHS

Abstract

In this paper, it has been shown that the hairy cycle C_n ⊙ rK₁, n ≡ 3(mod4), the graph obtained by adding pendant edge to each pendant vertex of hairy cycle C_n ⊙ 1K₁, n ≡ 0(mod4), double graph of path P_n and double graph of comb P_n ⊙ 1K₁ are k-graceful.

1. Introduction

The k-graceful labeling is the generalization of graceful labeling that intro-duced by Slater [14] in 1982 and by Maheo and Thuillier [10] also in 1982. Let G(V,E) be a simple undirected graph with order p and size q, k be an arbitrary natural number, if there exist an injective mapping f : V (G) → {0, 1,...,q+k−1} that induces bijective mapping f∗ : E(G) → {k, k + 1,...,q + k − 1} where f∗(u, v) = |f(u) − f(v)| ∀ (u, v) ∈ E(G) and u, v ∈ V (G) then f is called k-graceful labeling, while f∗ is called an induced edges k-graceful labeling and the graph G is called k-graceful graph. Graphs that are k-graceful for all k are sometimes called arbitrarily graceful.

Maheo and Thuillier [10] have shown that cycle Cn is k-graceful if and only if either n ≡ 0 or 1(mod4) with k even and k ≤ (n − 1)/2 or n ≡ 3(mod4) with k odd and k ≤ (n2 − 1)/2, while P. Pradhan and et al.[11] have shown that cycle Cn, n ≡ 0(mod4) is k-graceful for all k ∈ N(set of natural numbers). Maheo and Thuillier [10] have also proved that the wheel graph W2k+1 is k-graceful and conjecture that W2k is k-graceful when k ≠ 3 or k ≠ 4. This conjecture has proved by Liang, Sun and Xu [8]. Liang and Liu [7] have shown that Km,n is k-graceful. Acharya [1] has shown that eulerian graph with q edges is k-graceful if either q ≡ 0 or 1(mod4) with k even or q ≡ 3(mod4) with k odd. Seoud and Elsakhawi [12] have shown that paths and ladders are k-graceful.

Jirimutu [5] has shown that the graph obtained from K1,n(n ≥ 1) by attaching r ≥ 2 edges at each vertex is k-graceful for all k ≥ 2. After that Jirimutu, Bao and Kong [6] have shown that the graph obtained from K2,n(n ≥ 2) and K3,n(n ≥ 3) by attaching r ≥ 2 edges at each vertex is k-graceful for all k ≥ 2 and Siqinqimuge and Jirimutu [13] have proved that the graph obtained from K4,n(n ≥ 4) by attaching r ≥ 2 edges at each vertex is k-graceful for all k ≥ 2. Deligen, Zhao and Jirimutu [3] have proved that the graph obtained from K5,n(n ≥ 5) by attaching r ≥ 2 edges at each vertex is k-graceful for all k ≥ 2. Bu, Zhang and He [2] have shown that an even cycle with a fixed number of pendant edges adjoined to each vertex is k-graceful.

In the following section, it has been shown that the hairy cycle Cn⊙rK1, n ≡ 3(mod4) is k-graceful and the graph obtained by adding pendant edge to pendant vertex of hairy cycle Cn ⊙ 1K1, n ≡ 0(mod4) is also k-graceful.

 

2. Hairy Cycle

A unicycle graph other than a cycle with the property that the removal of any edge from the cycle reduces G to a caterpillar is called hairy cycle. The corona of cycle Cn and rK1 i.e. Cn ⊙ rK1 is the example of hairy cycle.

Theorem 2.1. The hairy cycle Cn ⊙ rK1, n ≡ 3(mod4) is admits k-graceful labeling where k ≤ r.

Proof. Let ui(i = 1, 2,...,n) be the cycle vertices of hairy cycle Cn⊙rK1 and the vertices of the r-hanged edges connected to each ui(i = 1, 2,...,n) are denoted by uit(t = 1, 2,...,r).

Consider the map f : V (Cn⊙rK1) → {0, 1,...,n(r+1)+k−1} defined as follows:

and

It is easy to check that f is injective mapping from V (Cn⊙rK1) to {0, 1,...,n(r+1) + k − 1}. Now we prove that the induced mapping f∗ : E(Cn ⊙ rK1) → {k, k + 1,...,n(r + 1) + k − 1} where f∗(u, v) = |f(u) − f(v)| is a bijective mapping for all edges (u, v) ∈ E(Cn ⊙ rK1). Let

The edge label induced by f∗ is as follows.

We tide up the elements of each set and have a union

So the induced mapping f∗ is a bijective mapping from V (Cn ⊙ rK1) onto {k, k + 1,...,n(r + 1) + k − 1}. Thus, the hairy cycle Cn ⊙ rK1, n ≡ 3(mod4) is admits k-graceful labeling. For example, 3-graceful labeling of hairy cycle C7 ⊙ 4K1, has shown in Fig. 1. □

Figure 1.3-graceful labeling of hairy cycle C7 ⊙ 4K1

Theorem 2.2. The graph obtained by adding pendant edge to each pendant vertex of hairy cycle Cn ⊙ 1K1, n ≡ 0(mod4) admits k-graceful labeling.

Proof. The order and size of the graph G obtained by adding pendant edge to each pendant vertex of hairy cycle Cn ⊙ 1K1, n ≡ 0(mod4) are respectively 3n and 3n. Let u1, u2,...,un be the cycle vertices of Cn ⊙ 1K1, v1, v2,...,vn be the vertices adjacent to u1, u2,...,un and w1,w2,...,wn be the vertices adjacent to 1K1, v1, v2,...,vn respectively. Obviously

Consider a labeling map f : V (G) → {0, 1,...,3n + k − 1} defined as follows:

It is clear that f is injective and the induced labeling map f∗ : E(G) → {k, k + 1,...,3n + k − 1} defined as f∗(u, v) = |f(u) − f(v)| ∀ (u, v) ∈ E(G) and u, v ∈ V (G), where u and v are adjacent vertices of G, is bijective. Thus f is k-graceful labeling of the graph G. For example, the graph obtained by adding pendant edge to each pendant vertex of C16 ⊙ 1K1 and its 3-graceful labeling are shown in Fig. 2 and Fig. 3 respectively. □

Figure 2

Figure 3

 

3. Double graph:

Let G′ be a copy of simple graph G, let ui be the vertices of G and vi be the vertices of G′ correspond with ui. A new graph denoted by D(G) is called the double graph of G[9] if

V (D(G)) = V (G) ∪ V (G′) and

E(D(G)) = E(G) ∪ E(G′) ∪{uivj : ui ∈ V (G), vj ∈ V (G′) and uiuj ∈ E(G)}

Theorem 3.1. Double graph of path Pn(n > 1) is k-graceful.

Proof. Let Pn be a path with n vertices u1, u2,...,un and vi be the copy of ui, then the path P′n = v1, v2,...,vn be copy of Pn. Double graph of path Pn denoted by D(Pn) have order and size 2n and 4(n−1) respectively. In the following Fig. 4, Fig. 5 and Fig. 6, we have shown path P9, P′9 and double graph D(P9) respectively.

Figure 4.Path P9

Figure 5.Path P′9

Figure 6.Double graph D(P9)

Consider the mapping f : V (D(Pn)) → {0, 1,...,4(n − 1) + k − 1} defined as follows:

It is clear that f is injective and the induced labeling map f∗ : E(D(Pn)) → {k, k + 1,...,4(n − 1) + k − 1} defined as f∗(u, v) = |f(u) − f(v)| ∀ (u, v) ∈ E(D(Pn)) and u, v ∈ V (D(Pn)), where u and v are adjacent vertices of D(Pn), is bijective. Thus f is k-graceful labeling of the double graph D(Pn). Hence the double graph D(Pn) is k-graceful. In the following Fig. 7, we have shown the 3-graceful labeling of the double graph D(P9).

Figure 7.3-graceful labeling of the double graph D(P9)

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Theorem 3.2. Double graph of comb graph Pn ⊙ 1K1(n > 1) is k-graceful.

Proof. Let {v1, v2,...,vn} be the set of path vertices and {u1, u2,...,un} be the set of pendant vertices of comb graph Pn ⊙ 1K1 such that vi is adjacent to ui, i = 1, 2,...,n. Similarly, let be the set of path vertices and be the set of pendant vertices of comb graph (Pn ⊙ 1K1)′ such that is adjacent to , i = 1, 2,...,n. Double graph of comb Pn ⊙1K1 denoted by D(Pn ⊙ 1K1) have order and size 4n and 4(2n − 1) respectively. In the following Fig. 8, and Fig. 9, we have shown comb graph P7 ⊙ 1K1 and double graph D(P7 ⊙ 1K1) respectively.

Figure 8.Comb graph P7 ⊙ 1K1

Figure 9.Double graph D(P7 ⊙ 1K1)

Consider the mapping f : V (D(Pn)) → {0, 1,...,4(2n−1)+k −1} defined as follows:

It is clear that f is injective and the induced labeling map f∗ : E(D(Pn ⊙ 1K1)) → {k, k+1,...,4(2n−1)+k−1} defined as f∗(u, v) = |f(u)−f(v)| ∀ (u, v) ∈ E(D(Pn ⊙ 1K1)) and u, v ∈ V (D(Pn ⊙ 1K1)), where u and v are adjacent ver-tices of D(Pn ⊙ 1K1), is bijective. Thus f is k-graceful labeling of the double graph D(Pn ⊙1K1). Hence the double graph D(Pn ⊙1K1) is k-graceful. In the following Fig. 10, we have shown the 4-graceful labeling of the double graph D(P7 ⊙ 1K1).

Figure 10.4-graceful labeling of the double graph D(P7 ⊙ 1K1)

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