International Journal of Pure and Applied Mathematics Volume 109 No. 7 2016, 159-166 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v109i7.20 ijpam.eu
SHELL BUTTERFLY GRAPHS ARE EDGE ODD GRACEFUL
J. Jeba Jesintha1 and K. Ezhilarasi Hilda2 1PG Department of Mathematics Women ’s Christian College, Chennai , INDIA 2 Department of Mathematics Ethiraj College for Women, Chennai , INDIA
Abstract: Let G be a graph with q edges. G is said to be edge odd graceful if there is a bijection f : E(G) → {1, 3, 5,..., (2q − 1)} such that , the induced mapping f ∗ : V (G) → {1, 2, 3,..., ∗ (2q − 1)} is given by f (x) = {P f(xy)}(mod 2q) and the resulting vertex labels are distinct. Shell graphs are the join of K1 and Pk , the path with ’k’ vertices. Shell butterfly graphs are one point union of two shells of any order and two pendant edges at the apex. In this paper we prove that Shell butterfly graphs are edge odd graceful.
AMS Subject Classification: 05C78 Key Words: Edge odd graceful labeling, shell graph , shell butterfly graph
1. Introduction
Most graph labeling methods trace their origin to the one introduced by Rosa [11] in the year 1967. He introduced the labeling method called β - valuation as a tool for decomposing the complete graph into isomorphic sub graphs . Later on this β - valuation was renamed as graceful labeling by Golomb [4]. Following the graceful labeling, dozens of other graph labeling techniques have been studied. One such labeling method is odd graceful labeling introduced by Gnanajothi [3] in the year 1991. A graph G with q edges is said to be an odd graceful graph if there is an injection f : V (G) → {0, 1, 2,..., (2q − 1)}
Received: October 1, 2016 c 2016 Academic Publications, Ltd. Published: April 25, 2016 url: www.acadpubl.eu 160 J. Jeba Jesintha, K. Ezhilarasi Hilda such that when each edge xy is assigned the label |f(x) − f(y)|, the resulting edge labels are {1, 3, 5, ..., (2q − 1)}. She has proved that the Paths Pn, cycles Cn( n-even), complete bipartite graphs, combs(the graph obtained by attaching pendant edge to each vertex of the path Pn(n > 1) ), books, the disjoint union of copies of C4 are all odd graceful. Lo [10], in the year 1985 introduced the notion of edge graceful graphs. A graph with n vertices and q edges is said to be edge graceful if there exists a bijection f : E(G) → {0, 1, 2, . . . , q} such that the induced labeling f ∗(x) = { f(xy)}(mod n) taken over all edges xy, is a bijection. P Motivated by the edge graceful labeling of Lo, a new type of labeling called an edge odd graceful labeling was introduced by Solairaju and Chitra [12] in the year 2008. A graph G with q edges is said to be edge odd graceful if there is a bijection f : E(G) → {1, 3, 5,..., (2q − 1)} such that, the induced mapping f ∗ : V (G) → {1, 2, 3,..., (2q − 1)} is given by f ∗(x) = { f(xy)}(mod 2q) and the resulting vertex labels are distinct. Solairaju and ChitraP In [13] proved the following graphs to be edge odd graceful: combs, the bi star, the tree < K1,n : 2 > and the graph obtained by subdividing each edge of the star K1,2n. For more results on edge odd graceful labeling refer [2].
Deb and Limaye [1] have defined a shell graph as a cycle Cn with (n - 3) chords sharing a common end point called the apex. In other words shell graphs are the join of K1 and Pk , the path with ’k’ vertices. Shell graphs are denoted as C(n, n- 3). A multiple shell is defined to be a collection of edge disjoint shells that have their apex in common. Hence a double shell consists of two edge disjoint shells with a common apex. In [5] a bow graph is defined as a double shell in which each shell has any order. A bow graph with exactly two pendant edges at the apex is known as a Shell butterfly graph. The shell butterfly graphs are proved to be graceful in [6], k-graceful in [7], harmonious in [8], and have ρ∗ valuation in [9]. In this paper we prove that all shell butterfly graphs are edge odd graceful.
2. Main Result
In this section we prove that all shell butterfly graphs are edge odd graceful.
Theorem 1. Shell butterfly graphs are edge odd graceful when ℓ = 2m + 1, ℓ = 2m + 3 and ℓ = 2m where ’m’ and ’ℓ’ are path orders of the shells. SHELL BUTTERFLY GRAPHS ARE EDGE ODD GRACEFUL 161
Proof. Let G be a shell butterfly graph with path orders ’m’ and ’ℓ’. We denote the apex of G as v0. Denote the vertices in the path of the right shell of G from bottom to top as v1, v2, . . . , vm. The vertices in the path of the left shell of G from top to bottom are denoted as vm+1, vm+2, . . . , vm+ℓ. The pendant vertices are denoted as vm+ℓ+1 and vm+ℓ+2. Let e1, e2, e3, . . . , eℓ−2, eℓ−1, eℓ be the edges joining v0 and the vertices vm+1, vm+2, . . . , vm+ℓ respectively. Let eℓ+1, eℓ+2, eℓ+3, . . . , em+ℓ−2, em+ℓ−1, em+ℓ be the edges joining v0 and the ver- tices vm, vm−1, vm−2 . . . , v3, v2, v1 respectively. Let em+ℓ+1, em+ℓ+2, em+ℓ+3, . . . , em+2ℓ−2, em+2ℓ−1 be the edges (vm+1, vm+2), (vm+2, vm+3), (vm+3, vm+4), ..., (vm+ℓ−2, vm+ℓ−1), (vm+ℓ−1, vm+ℓ) in the left shell. Let em+2ℓ, em+2ℓ+1,..., e2m+2ℓ−3, e2m+2ℓ−2 be the edges (vm, vm−1), (vm−1, vm−2),..., (v3, v2), (v2, v1) in the right shell. Let em+2ℓ−1 and e2m+2ℓ be the pendant edges (v0, v2m+2ℓ−1) and (v0, v2m+2ℓ) respectively. See Figure 1. Note that q = 2(m + l), n = m +l+3.
The edge labeling for the above described shell butterfly graph is defined as follows :-
f(ei) = 2i − 1, for 1 ≤ i ≤ q (1)
In equation (1) we can see that the edge labels are distinct odd numbers from
Figure 1: A shell butterfly graph 162 J. Jeba Jesintha, K. Ezhilarasi Hilda
1 to (2q -1).
The vertex labels are computed by the definition of edge odd graceful la- beling as follows:-
∗ f (v0) = {f(eq) + f(eq−1) + f(ei)}(mod 2q), for1 ≤ i ≤ (m − 1) (2)
∗ f (v1) = {f(em+1) + f(e2m+2ℓ−2)}(mod 2q) (3)
∗ f (vi) = {f(em+ℓ−i+1) + f(e2m+2ℓ−i) + f(e2m+2ℓ−i−1)}(mod 2q) (4)
for 2 ≤ i ≤ (m − l)
∗ f (vm) = {f(em+2ℓ) + f(eℓ+1)}(mod 2q) (5)
∗ f (vm+1) = {f(e1) + f(em+ℓ+1)}(mod 2q) (6)
∗ f (vi) = {f(ei+ℓ−1) + f(ei+ℓ) + f(ei−m)}(mod 2q) (7)
for(m + 2) ≤ i ≤ (m + ℓ − 1)
∗ f (vm+ℓ) = {f(eℓ) + f(em+2ℓ−1)}(mod 2q) (8)
∗ f (vm+ℓ+1) = {f(e2m+2ℓ))}(mod 2q) (9)
∗ f (vm+ℓ+2) = {f(e2m+2ℓ−1))}(mod 2q) (10)
The above vertex labels are shown to be distinct in the following three cases, namely ℓ = 2m + 1, ℓ = 2m + 3, ℓ = 2m where ℓ is the number of vertices in path of the left shell. SHELL BUTTERFLY GRAPHS ARE EDGE ODD GRACEFUL 163
Case 1 : When ℓ = (2m+1)
(6m − 4), for i = 1 (6m − 6i − 1), for 2 ≤ i ≤ (m − 1) (2m + 2), for i = m ∗ (6m + 4), for i = m + 1 f (vi) = (11) (6i − 6m − 5), for (m + 2) ≤ i ≤ (m + ℓ − 1) (2m − 2), for i = (m + 1) (2q − 1), for i = (m + ℓ + 1) (2q − 3), for i = (m + ℓ + 2)
(6m − 2), when m = 4k -1,k ≥ 1 ∗ (3m − 3), when m = 4k, k ≥ 1 f (v0) = (12) 12m, when m = 4k +1,k ≥ 1 (9m − 1), when m = 4k+2, k ≥ 1
From equations (11) and (12) it is clear that the vertex labels are all distinct. An illustration is given in Figure 2. Case 2 : When ℓ = (2m+3)
6m, for i = 1 (6m − 6i + 3), for 2 ≤ i ≤ (m − 1) (2m + 6), for i = m ∗ (6m + 8), for i = m + 1 f (vi) = (13) (6i − 6m − 5), for (m + 2) ≤ i ≤ (m + ℓ − 1) (2m + 2), for i = (m + 1) (2q − 1), for i = (m + ℓ + 1) (2q − 3), for i = (m + ℓ + 2)
(12m + 8), when m = 4k -1,k ≥ 1 ∗ (9m + 5), when m = 4k, k ≥ 1 f (v0) = (14) (6m + 2), when m = 4k +1,k ≥ 1 (3m − 1), when m = 4k+2, k ≥ 1 In this case also the vertices have distinct labels as seen in equations (13) and (14). Also f ∗(v) ⊆ {1, 2, 3,..., (2q − 1)}. 164 J. Jeba Jesintha, K. Ezhilarasi Hilda
Case 3 : When ℓ = 2m (6m − 6), for i = 1 (6m − 6i − 3), for 2 ≤ i ≤ (m − 1) 2m, for i = m ∗ (6m + 2), for i = m + 1 f (vi) = (15) (6i − 6m − 5), for (m + 2) ≤ i ≤ (m + ℓ − 1) (2m − 4), for i = (m + 1) (2q − 1), for i = (m + ℓ + 1) (2q − 3), for i = (m + ℓ + 2) (3m − 4), when m = 4k -1,k ≥ 1 ∗ (12m − 4), when m = 4k, k ≥ 1 f (v0) = (16) (9m − 4), when m = 4k +1,k ≥ 1 (6m − 4), when m = 4k+2, k ≥ 1
Figure 2: An edge odd graceful shell butterfly graph when m = 5, ℓ = 11, n = 19, q = 32 SHELL BUTTERFLY GRAPHS ARE EDGE ODD GRACEFUL 165
Here too as seen from equations (15) and (16) the vertices have distinct labels.
Thus in all the three cases the vertices have distinct labels and the edge labels are also distinct and odd. Hence shell butterfly graphs with path orders ’ m ’ and ’ℓ’ are edge odd graceful .
3. Acknowledgement
We wish to thank Dr. Indra Rajasingh and Dr. Kalyani Desikan for their help and support to publish this result in the International Journal of Pure and Applied Mathematics.
References [1] P. Deb and N.B. Limaye, On Harmonious Labeling of some cycle related graphs, Ars Combina. 65,(2002), 177-197. [2] J. A. Gallian , A Dynamic survey of Graph labeling , Elec .Journal of combina. (2010) November 13. [3] R. B.Gnanajothi, Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University, (1991). [4] S. W. Golomb, How to number a graph in Graph Theory and computing, R.C. Read,ed., Academic Press, New York ,(1972) 23- 37. [5] J. Jeba Jesintha and Ezhilarasi Hilda Stanley, Butterfly graphs with shell orders m and 2m+1 are graceful, Bon fring International Journal of Research in communication En- gineering,Vol.2(2) , (2012), 1-5 [6] J. Jeba Jesintha and K. Ezhilarasi Hilda, Shell Butterfly Graphs are Graceful, Interna- tional Journal of Pure and Applied Mathematics , Vol 101(5), (2015), 949 - 956. [7] J. Jeba Jesintha and K. Ezhilarasi Hilda, Double shell with two pendant edges at the apex are k-graceful, Journal of combinatorial Mathematics and Combinatorial Computing , Vol.92 , ISSN: 0835 3026, (2015), 47-57. [8] J. Jeba Jesintha and K. Ezhilarasi Hilda, Shell Butterfly Graphs are Harmonious, Ad- vances and Applications in Discrete Mathematics, Vol.14(1), (2014), 39-51. [9] J. Jeba Jesintha, and K. Ezhilarasi Hilda, ρ* labeling of Paths and Shell Butterfly Graphs, International Journal of Pure and Applied Mathematics, Vol.101(5), (2015), 645 - 653. [10] S. Lo, On edge graceful labeling of graphs, Congress Numer., (1985), 231-241. [11] A. Rosa, On certain valuations of the vertices of a graph ,Theory of graphs Proceedings of the symposium, Rome, Gordon and Breach, New York, (1967), 349-355. [12] A. Solairaju, and K. Chithra,Edge odd graceful labeling of some graphs, Proceedings of the ICMCS, 1 ,(2008), 101-107. 166 J. Jeba Jesintha, K. Ezhilarasi Hilda
[13] A. Solairaju and K. Chitra , Edge odd graceful graphs , Electronic Notes in Discrete Mathematics, 33 , (2009), 15-20.