 Open Access
 Total Downloads : 398
 Authors : A.Solairaju, N.Abdul Ali
 Paper ID : IJERTV1IS10067
 Volume & Issue : Volume 01, Issue 10 (December 2012)
 Published (First Online): 28122012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Gracefull Ness of P_{k} 2_{Ck}
Gracefull Ness Of Pk 2
A. SolairajuÂ¹ And N. Abdul AliÂ²
12: P.G. & Research Department of Mathematics, Jamal Mohamed College, Trichy 20.
Abstract: In this paper, we obtained that the connected graph Pk 2C4 is graceful.
Most graph labeling methods trace their origin to one introduced by Rosa [2] or one given Graham and Sloane [1]. Rosa defined a function f, a valuation of a graph with q edges if f is an injective map from the vertices of G to the set {0, 1, 2 ,,q} such that when each edge xy is assigned the label f(x)f(y), the resulting edge labels are distinct.
A. Solairaju and K. Chitra [3] first introduced the concept of edgeodd graceful labeling of graphs, and edgeodd graceful graphs.
A. Solairaju and others [5,6,7] proved the results that(1) the Gracefulness of a spanning tree of the graph of Cartesian product of Pm and Cn,was obtained (2) the Gracefulness of a spanning tree of the graph of cartesian product of Sm and Sn, was obtained (3) edgeodd Gracefulness of a spanning tree of Cartesian product of P2 and Cn was obtained (4) Even edge Gracefulness of the Graphs was obtained (5) ladder P2 x Pn is evenedge graceful, and (6) the evenedge gracefulness of Pn O nC5 is obtained.
Section I : Preliminaries
Definition 1.1: Let G = (V,E) be a simple graph with p vertices and q edges.
A map f :V(G) {0,1,2,,q} is called a graceful labeling if

f is one to one

The edges receive all the labels (numbers) from 1 to q where the label of an edge is the absolute value of the difference between the vertex labels at its ends.
A graph having a graceful labeling is called a graceful graph.
Example 1.1: The graph 6 P5 is a graceful graph.
Section II Path merging with circulits of length four
Definition 2.1: Pk 2C4 is a connected graph obtained by merging a circuit of length 4 with isolated vertex of a path of length k.
Theorem 2.1: The connected graph Pk 2C4 is graceful.
T2
VK+2
T1 T2
V1 V2 V3 V4 V5 V6 VK1 VK
VK+1
VK+4
T3 VK+3
Define f: V {1,, q} by
f(T1) = 0; f(T2) = q, f(T3) = q1, f(T4) = 2
f(V ) = (q2) (1), i is odd, i =1,3, , k+1
i 2
(2+ ), i is even, i = 2,4,, k+2
2
f(Vk+3) = f(Vk+2) + 1 f(Vk+4) = f(Vk+3) + 1
T2
VK+2
T1 T2
VK+1
VK+4
V1 V2 V3 V4 V5 V6
VK2
VK1 VK
T3 VK+3
Define f: V {1,, q}by
f(T1) = 0; f(T2) = q, f(T3) = q1, f(T4) = 2
f(V ) = (q2) (1), i is odd, i =1,3, , k, k+2
i 2
(2+ ), i is even, i = 2,4,, k+1
2
f(Vk+3) = f(Vk+2) – 1 f(Vk+4) = f(Vk+3) – 1
Example 2.1: k = 11 (odd) ; P: V  19; Q: e  20
12
20
18
20
16 15 14 13 12 11
0 2
10 09 08
07 06
04 02
05
8 10
19 17
18 3
17 4 16 5 15 6 14 7 13
03 01
19 11
Example 2.2: k =14 (even) ; P: V  22; Q: e  23
23
21
23
19 18 17 16 15 14
13 12 11
10 09 08
10
04 02
07 06 05
0 2
22 20
21 3
20 4 19 5 18 6 17 7 16
8 15 9
14 12
03 01
22 11
References:

R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graph, SIAM J. Alg. Discrete Math., 1 (1980) 382 404.

A. Rosa, On certain valuation of the vertices of a graph, Theory of graphs (International Synposium,Rome,July 1966),Gordon and Breach, N.Y.and Dunod Paris (1967), 349355.

A.Solairaju and K.Chitra Edgeodd graceful labeling of some graphs, Electronics Notes in
Discrete Mathematics Volume 33,April 2009,Pages 1.

A. Solairaju and P.Muruganantham, evenedge gracefulness of ladder, The Global Journal of Applied Mathematics & Mathematical Sciences(GJAMMS). Vol.1.No.2, (JulyDecember 2008):pp.149153.

A. Solairaju and P.Sarangapani, evenedge gracefulness of Pn O nC5, Preprint (Accepted for publication in Serials Publishers, New Delhi).

A.Solairaju, A.Sasikala, C.Vimala Gracefulness of a spanning tree of the graph of product of Pm and Cn, The Global Journal of Pure and Applied Mathematics of Mathematical Sciences, Vol. 1, No2 (JulyDec 2008): pp 133136.

A. Solairaju, C.Vimala,A.Sasikala Gracefulness of a spanning tree of the graph of Cartesian product of Sm and Sn, The Global Journal of Pure and Applied Mathematics of Mathematical Sciences, Vol. 1, No2 (JulyDec 2008): pp117120.