- Email: [email protected]
Some new families of Graceful Graphs
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Electronic Notes in Discrete Mathematics 48 (2015) 127–133 www.elsevier.com/locate/endm
Some new families of Graceful Graphs Samina Abbas Boxwala 1 Department of Mathematics,Nowrosjee Wadia College of Arts and Science Savitribai Phule Pune University Pune, India.
Priyam Vashishta
2
Department of Mathematics, RTCCS, Kharghar University of Mumbai Mumbai, India
Abstract In this paper we show that the graph obtained by switching an arbitrary vertex of a cycle, duplication of an arbitrary vertex on the rim of a wheel with even vertices and mirror graph of a path are graceful. Keywords: Graceful graphs, duplication of a vertex, switching a vertex in a graph, Mirror graph.
1
Introduction
In 1967, Rosa in [8] introduced the concept of β-labeling of a simple, finite, connected and undirected graph G = (V, E). Golomb in [10] subsequently 1 2
Email: sammy [email protected] Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2015.05.018 1571-0653/© 2015 Elsevier B.V. All rights reserved.
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called such a labeling graceful. We refer to [6] for a dynamic survey on graph labeling. In this paper we investigate graceful labeling of graphs obtained by switching an arbitrary vertex of a cycle, duplication of an arbitrary vertex on the rim of a wheel with even vertices and mirror graph of a path. We give a brief summary of definitions and relevant results which are useful for the present investigation. Definition 1.1 If the vertices of a graph are assigned real numbers subject to certain conditions then it is known as graph labeling. A function f is called a graceful labeling of a graph G having n vertices and q edges if f : V → {0, 1, . . . , q} is injective and the induced function f ∗ : E → {1, 2, . . . , q} defined as f ∗ (e = uv) = |f (u) − f (v)| is bijective. A graph which admits a graceful labeling is called a graceful graph. Definition 1.2 The wheel graph Wn is defined to be the join K1 + Cn . The vertex corresponding to K1 is known as apex vertex and vertices corresponding to the cycle are known as rim vertices while the edges corresponding to the cycle are known as rim edges. We continue to recognize the apex of the wheel as apex of respective graphs obtained from the wheel. Definition 1.3 A vertex switching Gv of a graph G is the graph obtained by taking a vertex v of G, removing all the edges incident on v and adding edges joining v to every other vertex which is not adjacent to v in G; all edges for which v is not an end vertex remain the same. Definition 1.4 The duplication of a vertex vk of a graph G produces a new graph G1 by adding a new vertex vk in such a way that N (vk ) = N (vk ) where N (vk ) denotes the set of vertices adjacent to vk and is called the Neighbours of vk . Definition 1.5 For a bipartite graph G with partite sets V1 and V2 , let G be a copy of G and V1 , V2 be copies of V1 , V2 respectively. The mirrror graph M (G) of G is obtained from G and G by joining each vertex of V2 to its corresponding vertex in V2 by an edge.
2
Main Results
Theorem 2.1 The graph obtained by switching of an arbitrary vertex in a cycle Cn , where n > 3 is graceful. Proof. Let v0 , v1 , . . . , vn−1 be the successive vertices of Cn and Gv denote the graph obtained by switching of a vertex v of Cn . Without loss of generality,
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let the switched vertex be v0 . We initiate a labeling from v0 . To define f : V → {0, 1, . . . , q = 2n − 5}, we consider the following cases. Case 1. n is even. f (v0 ) = 0, f (v1 ) = n − 2. ⎧ ⎨ 2n − i − 3, i even; For 2 ≤ i ≤ n − 1: f (vi ) = ⎩ i − 2, i odd. We prove that f is injective. Firstly, observe that except for v0 and v1 , all remaining n − 2 vertices receive odd labels. Hence it is enough to prove that f (vi ) = f (vj ), for i, j > 1, i = j. If both i and j are odd or both are even, the result is easy. If i is even and j is odd, then f (vi ) = f (vj ) implies that 2n − i − 3 = j − 2 implies that i + j + 1 = 2n. But i ≤ n − 2, j ≤ n − 1, hence i + j + 1 < 2n. Thus f is injective. Next we prove that f ∗ is surjective. Since there are q edges and q possible edge labels, it is enough to prove that f ∗ is injective. The edges v0 v2 , v0 v3 , . . . , v0 vn−2 and v1 v2 receive odd labels, whereas v2 v3 , v3 v4 , . . . , vn−2 vn−1 receive even labels. ⎧ ⎨ 2n − i − 3, i even, i ≤ n − 2; For 2 ≤ i ≤ n − 2, f ∗ (v0 vi ) = ⎩ i − 2, i odd, i ≤ n − 3. ∗ ∗ and f (v1 v2 ) = n − 3. If f (v0 vi ) = f (v0 vj ) , where i is even and j is odd. Then 2n − i − 3 = j − 2 implies i + j = 2n − 1 which is not possible as i + j ≤ 2n − 5. Likewise 2n − i − 3 = n − 3 and i − 2 = n − 3. Hence no two edges with an odd label will have the same label. Further, for 2 ≤ i ≤ n − 1, f ∗ (vi vi+1 ) = 2n − 2i − 2. Hence it is obvious that f ∗ is injective. Case 2. n is odd. f (v0 ) = 0, f (vn−1 ) = n − 3. For 1 ≤ ⎧ i ≤ n, i = n − 1: ⎨ n − i − 2, i even; f (vi ) = ⎩ n + i − 3, i odd. The proof in this case is similar. 2 Theorem 2.2 The graph obtained by duplicating an arbitrary vertex on the rim of a wheel having even number of vertices on the rim is graceful. Proof. Let v be the apex vertex and v1 , v2 , . . . , vn be the rim vertices of the wheel Wn , where n is even. Let G be the graph obtained by duplicating an arbitrary vertex on the rim of Wn . Without loss of generality, let this vertex be v1 and the newly added vertex be v1 . Define f : V → {0, 1, . . . , q = 2n + 3} as follows : f (v) = 0, f (v1 ) = 1, f (vn−1 ) = 2, and f (vn ) = 2n + 3.
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⎧ ⎨ 2n − i + 1, i is even; For 1 ≤ i ≤ n − 2 f (vi ) = ⎩ i + 2, i odd. We prove that f is injective. Firstly, observe that except for v0 and vn−1 , all remaining n vertices receive odd labels. Also, labels of v, v1 , vn−1 and vn are all distinct. Hence it is enough to prove that f (vi ) = f (vj ), for 1 ≤ i, j ≤ n − 2, i = j. If both i and j are odd or both are even, the result is easy. If i is even and j is odd, then f (vi ) = f (vj ) implies that 2n − i + 1 = j + 2 implies that i + j + 1 = 2n. But i ≤ n − 2, j ≤ n − 3, hence i + j + 1 < 2n. Thus f is injective. Next we prove that f ∗ is surjective. Since there are q edges and q possible edge labels, it is enough to prove that f ∗ is injective. The edges vv1 , vn−2 vn−1 , vn−1 vn , vv1 , . . . , vvn−2 and vvn receive odd labels whereas v1 v2 , v2 v3 , . . . , vn−3 vn−2 , vn v1 and vvn−1 receive even labels. By comparing the edge labels of any two edges with both edges having even labels or both having odd labels, it can be verified that f ∗ is injective. 2 Theorem 2.3 The mirror graph of a path is graceful. Proof. Let v0 , v1 , . . . , vn−1 be the vertices and e0 , e1 , . . . , en−2 be the edges of the path Pn such that vi is adjacent to vi+1 , 0 ≤ i ≤ n − 2. Pn is a bipartite graph. Let V1 = {v0 , v2 , . . .} and V2 = {v1 , v3 , . . .} be the bipartition of Pn . Let Pn be a copy of Pn and V1 = {v0 , v2 , . . .} and V2 = {v1 , v3 , . . .} be copies of V1 and V2 respectively. Let e0 , e1 , . . . , en−2 be copies of the edges of Pn . The mirror graph is obtained by joining the vertices of V2 to its corresponding vertices in V2 . Define f : V → {0, 1, . . . , q} as follows : Case 1. n ≡ 0(mod 4), where q = 5n−4 , f (v0 ) = q, f (v1 ) = 0. 2 ⎧ ⎨ 5n−4−i , i = 2, 4, . . . , n − 2; 2 For 2 ≤ i ≤ n − 1: f (vi ) = ⎩ f (v ) + 1, i = 3, 5, . . . , n − 1. i−2 ⎧ ⎪ ⎪ n − i − 2, i = 0, 2, 4, . . . , n−4 ; ⎪ 2 ⎨ When i is even, f (vi ) = 3n−6 , i = n2 ; 2 ⎪ ⎪ ⎪ ⎩ f (v ) − 2 i = n+4 , . . . , n − 2. i−2 2 When i is odd, f (vi ) = n + i − 1. Case 2. n ≡ 1(mod 4) and n = 5, where q = (5n−5) 2 f (v0 ) = q, f (v1 ) = 0 ⎧ ⎨ 5n−5−i , i = 2, 4, . . . , n − 1; 2 For 2 ≤ i ≤ n − 1: f (vi ) = ⎩ f (v ) + 1, i = 3, 5, . . . , n − 2. i−2
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f (vn−1 ) = 2n − 6, f (v0 ) = n + 1. ⎧ ⎪ ⎪ n − i, i = 2, 4, . . . , n−1 ; ⎪ 2 ⎨ When i is even, f (vi ) = n+7 , i = n+3 ; 2 2 ⎪ ⎪ ⎪ ⎩ f (v ) + 6, i = n+7 , . . . , n − 3. i−2 2 When i is odd, f (vi ) = n + i − 1. The case when n = 5 is to be dealt with separately and is labeled as shown in Figure 1 below.
8
1
7
4
9
0
5
10
3
6
Fig. 1. Graceful labeling of M (P5 ).
Case 3. n ≡ 2(mod 4), where q = (5n−4) . 2 f (v0 ) = q, f (v1 ) = 0 For 2 ≤ ⎧ i ≤ n − 1: ⎨ 5n−4−i , i = 2, 4, . . . , n − 2; 2 f (vi ) = ⎩ f (v ) + 1, i = 3, 5, , n − 1. i−2 f (v0 ) = n + 1, f (v2 ) = n − 2 When i ⎧ is even, ⎪ ⎪ n − i, i = 2, 4, . . . , n−2 ; ⎪ 2 ⎨ f (vi ) = n+4 , i = n+2 ; 2 2 ⎪ ⎪ ⎪ ⎩ 3i − n − 1, i = (n + 6)/2, . . . , n − 2. When i is odd, f (vi ) = n + i − 1. Case 4. n ≡ 3(mod 4) and n = 7, where q = f (v0 ) = q, f (v1 ) = 0. For 2 ≤ ⎧ i ≤ n − 1: ⎨ 5n−5−i , i = 2, 4, . . . , n − 1; 2 f (vi ) = ⎩ f (v ) + 1, i = 3, 5, . . . , n − 2. i−2 When i is even, f (vn−1 ) = 2n − 4
5n−5 2
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f (vi ) =
⎧ ⎪ ⎪ n − 2 − i, ⎪ ⎨ n+5
2 ⎪ ⎪ ⎪ ⎩ f (v
,
i = 0, 2, 4, . . . , n−3 ; 2 i=
n+1 ; 2
+ 6, i = n+5 , n+9 , . . . , n − 3. 2 2 When i is odd, f (vi ) = n + i − 1. The case when n = 7 is to be dealt with separately and the graph is labeled as shown in Figure 2. i−2 )
12
2
11
13
1
9
10
6
7
14
0
15
3
5
Fig. 2. Graceful labeling of M (P7 ).
2
3
Conclusion
In this paper we have proved that the graph obtained by switching an arbitrary vertex of a cycle, the graph obtained by duplication of an arbitrary vertex on the rim of a wheel with even vertices and the mirror graph of a path are graceful.
References [1] G.S. Bloom and S.W. Golomb, Applications of numbered undirected graphs, Proceedings of IEEE, 65(4) (1977), 562–570. [2] B. D. Acharya, Construction of certain infinite families of Graceful graphs from a given Graceful graph, Def. Sci.J., 32(3) (1982), 231–236. [3] Christian Barrientos, Graceful labeling of Chains and Corona graphs, Bull. Inst.Combin.Appli., 34 (2002), 17–26.
S.A. Boxwala, P. Vashishta / Electronic Notes in Discrete Mathematics 48 (2015) 127–133
133
[4] Christian Barrientos, Graceful graphs with pendant edges,Australian J. Combin., 33 (2005), 99–107. [5] D. Jin, S. Liu, S. Lee, H. Liu, X. Lu and D. Zhang, The joint sum of Graceful trees, Computers & Mathematics with Applications, 26(10) (1993), 83–87. [6] J. A. Gallian, A dynamic survey of graph labeling, The Electron. J. Combin., 18 (2011), #DS6. [7] J. Gross and J. Yellen, Graph Theory and its applications, CRC Press, 1999. [8] Rosa A, On certain valuations of the verices of a graph, In: Theory of Graphs, International Symposium, Rome, July 1966. [9] S. K. Vaidya, V. J. Kaneria, S. Srivastav and N. A. Dani, Gracefulness of union of two path graphs with grid graph and complete bipartite graph, Proceedings of the First International Conference on Emerging Technologies and Applications in Engineering, Technology and Sciences,(2008), 616–619. [10] S. W. Golomb, How to number a graph in Graph Theory and Computing, R C Read, ed., Academic Press, New York, (1972), 23–37.
Electronic Notes in Discrete Mathematics 48 (2015) 127–133 www.elsevier.com/locate/endm
Some new families of Graceful Graphs Samina Abbas Boxwala 1 Department of Mathematics,Nowrosjee Wadia College of Arts and Science Savitribai Phule Pune University Pune, India.
Priyam Vashishta
2
Department of Mathematics, RTCCS, Kharghar University of Mumbai Mumbai, India
Abstract In this paper we show that the graph obtained by switching an arbitrary vertex of a cycle, duplication of an arbitrary vertex on the rim of a wheel with even vertices and mirror graph of a path are graceful. Keywords: Graceful graphs, duplication of a vertex, switching a vertex in a graph, Mirror graph.
1
Introduction
In 1967, Rosa in [8] introduced the concept of β-labeling of a simple, finite, connected and undirected graph G = (V, E). Golomb in [10] subsequently 1 2
Email: sammy [email protected] Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2015.05.018 1571-0653/© 2015 Elsevier B.V. All rights reserved.
128
S.A. Boxwala, P. Vashishta / Electronic Notes in Discrete Mathematics 48 (2015) 127–133
called such a labeling graceful. We refer to [6] for a dynamic survey on graph labeling. In this paper we investigate graceful labeling of graphs obtained by switching an arbitrary vertex of a cycle, duplication of an arbitrary vertex on the rim of a wheel with even vertices and mirror graph of a path. We give a brief summary of definitions and relevant results which are useful for the present investigation. Definition 1.1 If the vertices of a graph are assigned real numbers subject to certain conditions then it is known as graph labeling. A function f is called a graceful labeling of a graph G having n vertices and q edges if f : V → {0, 1, . . . , q} is injective and the induced function f ∗ : E → {1, 2, . . . , q} defined as f ∗ (e = uv) = |f (u) − f (v)| is bijective. A graph which admits a graceful labeling is called a graceful graph. Definition 1.2 The wheel graph Wn is defined to be the join K1 + Cn . The vertex corresponding to K1 is known as apex vertex and vertices corresponding to the cycle are known as rim vertices while the edges corresponding to the cycle are known as rim edges. We continue to recognize the apex of the wheel as apex of respective graphs obtained from the wheel. Definition 1.3 A vertex switching Gv of a graph G is the graph obtained by taking a vertex v of G, removing all the edges incident on v and adding edges joining v to every other vertex which is not adjacent to v in G; all edges for which v is not an end vertex remain the same. Definition 1.4 The duplication of a vertex vk of a graph G produces a new graph G1 by adding a new vertex vk in such a way that N (vk ) = N (vk ) where N (vk ) denotes the set of vertices adjacent to vk and is called the Neighbours of vk . Definition 1.5 For a bipartite graph G with partite sets V1 and V2 , let G be a copy of G and V1 , V2 be copies of V1 , V2 respectively. The mirrror graph M (G) of G is obtained from G and G by joining each vertex of V2 to its corresponding vertex in V2 by an edge.
2
Main Results
Theorem 2.1 The graph obtained by switching of an arbitrary vertex in a cycle Cn , where n > 3 is graceful. Proof. Let v0 , v1 , . . . , vn−1 be the successive vertices of Cn and Gv denote the graph obtained by switching of a vertex v of Cn . Without loss of generality,
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let the switched vertex be v0 . We initiate a labeling from v0 . To define f : V → {0, 1, . . . , q = 2n − 5}, we consider the following cases. Case 1. n is even. f (v0 ) = 0, f (v1 ) = n − 2. ⎧ ⎨ 2n − i − 3, i even; For 2 ≤ i ≤ n − 1: f (vi ) = ⎩ i − 2, i odd. We prove that f is injective. Firstly, observe that except for v0 and v1 , all remaining n − 2 vertices receive odd labels. Hence it is enough to prove that f (vi ) = f (vj ), for i, j > 1, i = j. If both i and j are odd or both are even, the result is easy. If i is even and j is odd, then f (vi ) = f (vj ) implies that 2n − i − 3 = j − 2 implies that i + j + 1 = 2n. But i ≤ n − 2, j ≤ n − 1, hence i + j + 1 < 2n. Thus f is injective. Next we prove that f ∗ is surjective. Since there are q edges and q possible edge labels, it is enough to prove that f ∗ is injective. The edges v0 v2 , v0 v3 , . . . , v0 vn−2 and v1 v2 receive odd labels, whereas v2 v3 , v3 v4 , . . . , vn−2 vn−1 receive even labels. ⎧ ⎨ 2n − i − 3, i even, i ≤ n − 2; For 2 ≤ i ≤ n − 2, f ∗ (v0 vi ) = ⎩ i − 2, i odd, i ≤ n − 3. ∗ ∗ and f (v1 v2 ) = n − 3. If f (v0 vi ) = f (v0 vj ) , where i is even and j is odd. Then 2n − i − 3 = j − 2 implies i + j = 2n − 1 which is not possible as i + j ≤ 2n − 5. Likewise 2n − i − 3 = n − 3 and i − 2 = n − 3. Hence no two edges with an odd label will have the same label. Further, for 2 ≤ i ≤ n − 1, f ∗ (vi vi+1 ) = 2n − 2i − 2. Hence it is obvious that f ∗ is injective. Case 2. n is odd. f (v0 ) = 0, f (vn−1 ) = n − 3. For 1 ≤ ⎧ i ≤ n, i = n − 1: ⎨ n − i − 2, i even; f (vi ) = ⎩ n + i − 3, i odd. The proof in this case is similar. 2 Theorem 2.2 The graph obtained by duplicating an arbitrary vertex on the rim of a wheel having even number of vertices on the rim is graceful. Proof. Let v be the apex vertex and v1 , v2 , . . . , vn be the rim vertices of the wheel Wn , where n is even. Let G be the graph obtained by duplicating an arbitrary vertex on the rim of Wn . Without loss of generality, let this vertex be v1 and the newly added vertex be v1 . Define f : V → {0, 1, . . . , q = 2n + 3} as follows : f (v) = 0, f (v1 ) = 1, f (vn−1 ) = 2, and f (vn ) = 2n + 3.
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⎧ ⎨ 2n − i + 1, i is even; For 1 ≤ i ≤ n − 2 f (vi ) = ⎩ i + 2, i odd. We prove that f is injective. Firstly, observe that except for v0 and vn−1 , all remaining n vertices receive odd labels. Also, labels of v, v1 , vn−1 and vn are all distinct. Hence it is enough to prove that f (vi ) = f (vj ), for 1 ≤ i, j ≤ n − 2, i = j. If both i and j are odd or both are even, the result is easy. If i is even and j is odd, then f (vi ) = f (vj ) implies that 2n − i + 1 = j + 2 implies that i + j + 1 = 2n. But i ≤ n − 2, j ≤ n − 3, hence i + j + 1 < 2n. Thus f is injective. Next we prove that f ∗ is surjective. Since there are q edges and q possible edge labels, it is enough to prove that f ∗ is injective. The edges vv1 , vn−2 vn−1 , vn−1 vn , vv1 , . . . , vvn−2 and vvn receive odd labels whereas v1 v2 , v2 v3 , . . . , vn−3 vn−2 , vn v1 and vvn−1 receive even labels. By comparing the edge labels of any two edges with both edges having even labels or both having odd labels, it can be verified that f ∗ is injective. 2 Theorem 2.3 The mirror graph of a path is graceful. Proof. Let v0 , v1 , . . . , vn−1 be the vertices and e0 , e1 , . . . , en−2 be the edges of the path Pn such that vi is adjacent to vi+1 , 0 ≤ i ≤ n − 2. Pn is a bipartite graph. Let V1 = {v0 , v2 , . . .} and V2 = {v1 , v3 , . . .} be the bipartition of Pn . Let Pn be a copy of Pn and V1 = {v0 , v2 , . . .} and V2 = {v1 , v3 , . . .} be copies of V1 and V2 respectively. Let e0 , e1 , . . . , en−2 be copies of the edges of Pn . The mirror graph is obtained by joining the vertices of V2 to its corresponding vertices in V2 . Define f : V → {0, 1, . . . , q} as follows : Case 1. n ≡ 0(mod 4), where q = 5n−4 , f (v0 ) = q, f (v1 ) = 0. 2 ⎧ ⎨ 5n−4−i , i = 2, 4, . . . , n − 2; 2 For 2 ≤ i ≤ n − 1: f (vi ) = ⎩ f (v ) + 1, i = 3, 5, . . . , n − 1. i−2 ⎧ ⎪ ⎪ n − i − 2, i = 0, 2, 4, . . . , n−4 ; ⎪ 2 ⎨ When i is even, f (vi ) = 3n−6 , i = n2 ; 2 ⎪ ⎪ ⎪ ⎩ f (v ) − 2 i = n+4 , . . . , n − 2. i−2 2 When i is odd, f (vi ) = n + i − 1. Case 2. n ≡ 1(mod 4) and n = 5, where q = (5n−5) 2 f (v0 ) = q, f (v1 ) = 0 ⎧ ⎨ 5n−5−i , i = 2, 4, . . . , n − 1; 2 For 2 ≤ i ≤ n − 1: f (vi ) = ⎩ f (v ) + 1, i = 3, 5, . . . , n − 2. i−2
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f (vn−1 ) = 2n − 6, f (v0 ) = n + 1. ⎧ ⎪ ⎪ n − i, i = 2, 4, . . . , n−1 ; ⎪ 2 ⎨ When i is even, f (vi ) = n+7 , i = n+3 ; 2 2 ⎪ ⎪ ⎪ ⎩ f (v ) + 6, i = n+7 , . . . , n − 3. i−2 2 When i is odd, f (vi ) = n + i − 1. The case when n = 5 is to be dealt with separately and is labeled as shown in Figure 1 below.
8
1
7
4
9
0
5
10
3
6
Fig. 1. Graceful labeling of M (P5 ).
Case 3. n ≡ 2(mod 4), where q = (5n−4) . 2 f (v0 ) = q, f (v1 ) = 0 For 2 ≤ ⎧ i ≤ n − 1: ⎨ 5n−4−i , i = 2, 4, . . . , n − 2; 2 f (vi ) = ⎩ f (v ) + 1, i = 3, 5, , n − 1. i−2 f (v0 ) = n + 1, f (v2 ) = n − 2 When i ⎧ is even, ⎪ ⎪ n − i, i = 2, 4, . . . , n−2 ; ⎪ 2 ⎨ f (vi ) = n+4 , i = n+2 ; 2 2 ⎪ ⎪ ⎪ ⎩ 3i − n − 1, i = (n + 6)/2, . . . , n − 2. When i is odd, f (vi ) = n + i − 1. Case 4. n ≡ 3(mod 4) and n = 7, where q = f (v0 ) = q, f (v1 ) = 0. For 2 ≤ ⎧ i ≤ n − 1: ⎨ 5n−5−i , i = 2, 4, . . . , n − 1; 2 f (vi ) = ⎩ f (v ) + 1, i = 3, 5, . . . , n − 2. i−2 When i is even, f (vn−1 ) = 2n − 4
5n−5 2
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f (vi ) =
⎧ ⎪ ⎪ n − 2 − i, ⎪ ⎨ n+5
2 ⎪ ⎪ ⎪ ⎩ f (v
,
i = 0, 2, 4, . . . , n−3 ; 2 i=
n+1 ; 2
+ 6, i = n+5 , n+9 , . . . , n − 3. 2 2 When i is odd, f (vi ) = n + i − 1. The case when n = 7 is to be dealt with separately and the graph is labeled as shown in Figure 2. i−2 )
12
2
11
13
1
9
10
6
7
14
0
15
3
5
Fig. 2. Graceful labeling of M (P7 ).
2
3
Conclusion
In this paper we have proved that the graph obtained by switching an arbitrary vertex of a cycle, the graph obtained by duplication of an arbitrary vertex on the rim of a wheel with even vertices and the mirror graph of a path are graceful.
References [1] G.S. Bloom and S.W. Golomb, Applications of numbered undirected graphs, Proceedings of IEEE, 65(4) (1977), 562–570. [2] B. D. Acharya, Construction of certain infinite families of Graceful graphs from a given Graceful graph, Def. Sci.J., 32(3) (1982), 231–236. [3] Christian Barrientos, Graceful labeling of Chains and Corona graphs, Bull. Inst.Combin.Appli., 34 (2002), 17–26.
S.A. Boxwala, P. Vashishta / Electronic Notes in Discrete Mathematics 48 (2015) 127–133
133
[4] Christian Barrientos, Graceful graphs with pendant edges,Australian J. Combin., 33 (2005), 99–107. [5] D. Jin, S. Liu, S. Lee, H. Liu, X. Lu and D. Zhang, The joint sum of Graceful trees, Computers & Mathematics with Applications, 26(10) (1993), 83–87. [6] J. A. Gallian, A dynamic survey of graph labeling, The Electron. J. Combin., 18 (2011), #DS6. [7] J. Gross and J. Yellen, Graph Theory and its applications, CRC Press, 1999. [8] Rosa A, On certain valuations of the verices of a graph, In: Theory of Graphs, International Symposium, Rome, July 1966. [9] S. K. Vaidya, V. J. Kaneria, S. Srivastav and N. A. Dani, Gracefulness of union of two path graphs with grid graph and complete bipartite graph, Proceedings of the First International Conference on Emerging Technologies and Applications in Engineering, Technology and Sciences,(2008), 616–619. [10] S. W. Golomb, How to number a graph in Graph Theory and Computing, R C Read, ed., Academic Press, New York, (1972), 23–37.