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Strong edge graceful labeling of some graphs.

1. INTRODUCTION

All graphs in this paper are finite, simple and undirected. Terms not defined here are used in the sense of Harary [1]. The symbols V(G) will denote the vertex set and edge set of a graph G. The cardinality of the vertex set is called the order of G. The cardinality of the edge set is called the size of G. A graph with p vertices and q edges is called a (p,q) graph.

Lo [3] introduced the notion of edge graceful graphs. A graph G with q edges and p vertices is said to be edge graceful if there exists a bijection f from the edge set to the set {1,2, ..., q} so that the induced mapping [f.sup.+] from the vertex set to the set {0,1,2, ..., p-1} given by [f.sup.+](x) = [summation]{f(xy)/xy [member of]E(G)} (mod p) is a bijection.

The necessary condition for a graph to be edge graceful is q(q+1) [equivalent to] 0 or p/2 (mod p). With this condition one can verify that even cycles, and paths of even length are not edge graceful. But whether trees of odd order are edge graceful is still open. On these lines, we define a new type of labeling called strong edge graceful labeling by relaxing its range through which we can get edge graceful labeling of odd order trees for some family of graphs.

A (p,q) graph G is said to have strong edge graceful labeling if there exists an injection f from the edge set to {1, 2, ..., [3q/2]} so that the induced mapping [f.sup.+] from the vertex set to {0,1, ..., 2 p-1} defined by [f.sup.+](x) = [summation]{f(xy)}/xy [member of]E(G)} (mod 2p) are distinct. A graph G is said to be strong edge graceful if it admits a strong edge graceful labeling. In this paper, we investigate strong edge graceful labeling (SEGL) of some graphs.

2. MAIN RESULTS

Theorem 2.1: [C.sub.n] is strong edge graceful for all n where n is odd and n [greater than or equal to] 3.

Proof: Let [v.sub.1], [v.sub.2], ..., [v.sub.n] be the vertices of [C.sub.n] and [e.sub.1], [e.sub.2], ..., [e.sub.n] be the edges of [C.sub.n] denoted as in fig.1

[FIGURE 1 OMITTED]

We first label the edges [C.sub.n] of as follows

f ([e.sub.i]) = i for i = 1 to n. Then the induced vertex labels are

[f.sup.+] ([v.sub.1]) = n+1; [f.sup.+] ([v.sub.i]) = 2i-1 for i = 1,2,3, ..., n

Clearly {[f.sup.+] ([v.sub.i]): i = 1 to n} are distinct. Hence, [C.sub.n] is strong edge graceful for all n odd and n [greater than or equal to] 3. The strong edge graceful labeling of [C.sub.7] and [C.sub.11] are given below.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Theorem 2.2: [C.sub.n] is strong edge graceful for all n where n is even and n [greater than or equal to] 4.

Proof: Let [v.sub.1], [v.sub.2], ..., [v.sub.n] be the vertices of [C.sub.n] and [e.sub.1], [e.sub.2], ..., [e.sub.n] be the edges of [C.sub.n] denoted as in fig. 1. We first label the edges of [C.sub.n] as follows

f([e.sub.i]) = i for i = 1 to n-1 ; f([e.sub.n]) = n+1. Then the induced vertex labels are

[f.sup.+] ([v.sub.1]) n + 2; [f.sup.+] ([v.sub.n]) = 0

[f.sup.+] ([v.sub.2]) = 3; [f.sup.+] ([v.sub.i]) = [f.sup.+] ([v.sub.i-1]) + 2 for i = 3 to n-1

Clearly {[f.sup.+] ([v.sub.i]: i = 1 to no} are distinct. Hence, [C.sub.n] is strong edge graceful for all n even and n [greater than or equal to] 4. The strong edge graceful labeling of [C.sub.6] and [C.sub.8] are illustrated in fig. 4 and fig.5 respectively.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Theorem 2.3: [P.sub.n] is strong edge graceful for all n odd and n [greater than or equal to] 3.

Proof: Let [v.sub.1], [v.sub.2], ..., [v.sub.n] be the vertices of [P.sub.n] and {[e.sub.1], [e.sub.2], ..., [e.sub.n-1] be the edges of [P.sub.n] as defined in the fig. 6.

[FIGURE 6 OMITTED]

We first label the edges of [P.sub.n] as f([e.sub.i]) = i for i = 1 to n-1 Then the induced vertex labels are [f.sup.+] ([v.sub.1]) = 1; [f.sup.1] ([v.sub.1]) = 2i-1 for i = 2 to n-1.; [f.sup.+] ([v.sub.n]) = n-1

Clearly {[f.sub.+]([v.sub.i]): i = 1 to n} are all distinct. Hence [P.sub.n] is strong edge graceful for all n odd and n [greater than or equal to] 3. The strong edge graceful labeling of [P.sub.9] and [P.sub.13] are illustrated fig. 7 and fig. 8 respectively.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Theorem 2.4: [P.sub.n] is strong edge graceful for all n even and n [greater than or equal to] 4.

Proof: Let [v.sub.1], [v.sub.2] ..., [v.sub.n] be the vertices of [P.sub.n] and {[e.sub.1], [e.sub.2], ..., [e.sub.n-1]} be the edges of [P.sub.n] as defined in the fig. 6. We first label the edges of [P.sub.n] as follows:

f ([e.sub.i]) = i for i = 1 to n - 2; f([e.sub.n-1]) = n Then the induced vertex labels are

[f.sup.+] ([v.sub.i]) = 2i - 1 for i = 1 to n-2.; [f.sup.+] ([v.sub.n-i]) = 2n-2; [f.sup.+] ([v.sub.n]) = n. Clearly

{[f.sup.+] ([v.sub.i]): i = 1 to n} are all distinct. Hence [P.sub.n] is strong edge graceful for all n even and n [greater than or equal to] 4. The strong edge graceful labeling of [P.sub.6] and [P.sub.8] are illustrated fig. 9 and fig. 10 respectively.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Theorem 2.5: [C.sup.+.sub.n] is strong edge graceful for all n [greater than or equal to] 3.

Proof: Let {[v.sub.1], [v.sub.2] ..., [v.sub.n]} [union] {[v'.sub.1], [v'.sub.2] ..., [v'.sub.n]} be the vertices of [C.sup.+.sub.n]. The edges of [C.sup.+.sub.n] are {[e.sub.i] = [v.sub.i] [v.sub.i+1]: i = 1 to n - 1} [union] {[e.sub.n] = [v.sub.n] [v.sub.1]} [union] {[v.sub.i] [v'.sub.i]: i = 1 to n} are denoted as in fig.11

[FIGURE 11 OMITTED]

We first label the edges of [C.sup.+.sub.n] as follows: f([e.sub.i]) = i for i = 1 to n; f([v.sub.i] [v'.sub.i]) = 2n- i+1 for i = 1 to n

Then the induced vertex labels are

[f.sup.+]([v.sub.i]) = 2n + 1 + i for i = 1 to n. ; [f.sup.+]([v'.sub.i]) = 2n + 1 - i for i = 1 to n.

Clearly {[f.sup.+]([v.sub.i]), [f.sup.+]([v'.sub.i])} are distinct. Hence, [C.sup.+.sub.n] is strong edge graceful for all n [greater than or equal to] 3.

The labeling of [C.sup.+.sub.7] are [C.sup.+.sub.8] illustrated in fig. 12 and fig. 13 respectively.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

Theorem 2.6: [K.sub.1,2n] is strong edge graceful for all n [greater than or equal to] 1.

Proof: Let {v, [v.sub.1], [v.sub.2], ..., [v.sub.2n]} be the vertices of [K.sub.1,2n] and {[e.sub.1], [e.sub.2], [e.sub.3], ..., [e.sub.2n]} be the edges of [K.sub.1,2n] denoted as in fig. 14.

[FIGURE 14 OMITTED]

We first label the edges of [K.sub.1,2n] as f([e.sub.i]) = i for i = 1, 2, ..., 2n. Then the induced vertex labels are [f.sup.+]([v.sub.i]) = i for i = 1, 2, ..., 2n

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the induced vertex labels are all distinct. Hence, [K.sub.1,2n] is strong edge graceful for all n [greater than or equal to] 1. The strong edge graceful labeling of [K.sub.1,6] and [K.sub.1,8] are illustrated in fig.15 and fig 16 respectively.

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

Theorem 2.7: [K.sub.1,4n-1] is strong edge graceful for all n [greater than or equal to] 1.

Proof: Let {v, [v.sub.1], [v.sub.2], ..., [v.sub.4n-1]} be the vertices of [K.sub.1,4n-1] and {[e.sub.1], [e.sub.2], [e.sub.3], ..., [e.sub.4n-1]} be the edges of denoted as in fig.14. We first label the edges of [K.sub.1,4n-1] as follows f([e.sub.i]) = i for i = 1, 2, ..., 4n-1. Then the induced vertex labels are [f.sup.+]([v.sub.i]) = i for i = 1, 2, ..., 4n-1; [f.sup.+](v) = 6n. Then the induced vertex labels are all distinct. Hence, [K.sub.1,4n-1] is strong edge graceful for all n [greater than or equal to] 1. The strong edge graceful labeling of [K.sub.1,7] and [K.sub.1,11] are illustrated in fig.17 and fig 18 respectively.

[FIGURE 17 OMITTED]

[FIGURE 18 OMITTED]

Theorem 2.8: [K.sub.1,4n-1] is strong edge graceful for all n [greater than or equal to] 1.

Proof: Let {v, [v.sub.1], [v.sub.2], ..., [v.sub.4n-1]} be the vertices of [K.sub.1,4n-1] and {[e.sub.1], [e.sub.2], [e.sub.3], ..., [e.sub.4n-1]} be the edges of [K.sub.1,4n-1] denoted as in fig.14. We first label the edges of as follows f([e.sub.i]) = i for i = 1, 2, ..., 4n ; f([e.sub.4n+1]) = [3q/2] Then the induced vertex labels are [f.sup.+]([v.sub.i]) = i for i = 1, 2, ..., 4n - 1 ; [f.sup.+]([v.sub.4n+1]) = [3q/2] ; [f.sup.+](v) = 4n + 1 Then the induced vertex labels are all distinct. Hence, [K.sub.1,4n-1] is strong edge graceful for all n [greater than or equal to] 1. The strong edge graceful labeling of [K.sub.1,5] and [K.sub.1,9] are illustrated in fig.19 and fig. 20 respectively.

[FIGURE 19 OMITTED]

[FIGURE 20 OMITTED]

Definition 2.1: Let [P.sub.n] denote the path with n vertices. Then the join of [K.sub.1] with [P.sub.n] is defined as fan and is denoted by [F.sub.n] i.e. [F.sub.n] = [K.sub.1] + [P.sub.n].

Theorem 2.9: [F.sub.4n-2] is strong edge graceful graph for all n [greater than or equal to] 1.

Proof : Let {v, [v.sub.1], [v.sub.2], [v.sub.3], ..., [v.sub.4n-2]} be the vertices of [F.sub.4n-2] and the edges of [F.sub.4n-2] are defined as follows as denoted in fig. 21.

[FIGURE 21 OMITTED]

We first label the edges of [F.sub.4n-2] as follows: f([e.sub.i]) = i for i = 1 to 4n - 3, f(v[v.sub.i]) = 2n - i for i = 1 to 4n - 2 then the induced vertex labels are [f.sup.+]([v.sub.1]) = 2(4n - 2); [f.sup.+]([v.sub.i]) = f([v.sub.i-1]) + 1 for i = 2 to 4n - 1.; [f.sup.+]([v.sub.n-2]) = 8n - 5; [f.sup.+](v) = 4n + 1

The labeling of [K.sub.6], [K.sub.10] are illustrated in fig.22 and fig.23 respectively.

[FIGURE 22 OMITTED]

[FIGURE 23 OMITTED]

Theorem 2.10: [F.sub.4n+1] is strong edge graceful graph for all n [greater than or equal to] 1.

Proof Let {v, [v.sub.1], [v.sub.2], [v.sub.3], ..., [v.sub.4n+1]} be the vertices of [F.sub.4n+1] and the edges of [F.sub.4n+1] are defined as follows {v[v.sub.i]: i = 1 to 4n + 1}[union] {[e.sub.i] = ([v.sub.i],[v.sub.i+1]): i = 1 to 4n} as denoted in fig21. We first label the edges of [F.sub.4n+1] as follows: f([v.sub.i]) = i for i = 1 to 4n; f(v[v.sub.1]) = (4n + 1) f(v[v.sub.i]) = p (4n + 1) - i for i = 2 to (4n + 1) then the induced vertex labels are [f.sup.+]([v.sub.1]) = 4n + 2 ; [f.sup.+]([v.sub.i]) = n - 2 for i = 2 to 4n. [f.sup.+]([v.sub.4n+1]) = 2n ; [f.sup.+](v) = 3n + 7/2

The labeling of [F.sub.5] are illustrated in fig.24..

[FIGURE 24 OMITTED]

REFERENCES

[1.] Harary, F. 1972. Graph Theory. Addison Wesley, Mass Reading (1972).

[2.] Joseph, A. Gallian. 2007.A Dynamic Survey of Graph Labeling. The Electronic journal of combinatorics 14(2007)#DS6.

[3.] Lo, S. 1985. On edge graceful labeling of graphs, Congressus numerantium 50(1985).

[4.] Slamet, S. and Sugeng, K.A. 2008. Sharing scheme using magic covering--Preprint. 2008.

B. Gayathri * and M. Subbiah@

PG and Research Dept. of Mathematics, Periyar E.V.R. College, Trichy--23, India Email; maduraigayathri@gmail.com, mdthrcp@gmail.com
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Author:Gayathri, B.; Subbiah, M.
Publication:Bulletin of Pure & Applied Sciences-Mathematics
Article Type:Report
Geographic Code:9INDI
Date:Jan 1, 2008
Words:2263
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