ISSN ONLINE(2319-8753)PRINT(2347-6710)
K.Murugan* Assistant Professor, Department of Mathematics, The M.D.T. Hindu College, Tirunelveli, Tamilnadu, India |
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A ð. ð graph G = ð, ð¸ is said to be a square graceful graph ifthere exists an injective function f: V ðº → 0,1,2,3, … , ð2 such that the induced mapping ðð : E ðº → 1,4,9, … , ð2 defined by ðð ð¢ð£ = ð ð¢ − ð ð£ is an injection. The function f is called a square graceful labeling of G. In this paper the square graceful labeling of the caterpillar S ð1, ð2, … , ðð , the graphs ðð −1 1,2, … ð ,mð¾1,ð ∪ ð ð¾1,ð¡ , ð¾1,ð , ðð =1 ðð â¨ð¾1 − ð,H graph and some other graphsare studied. A new parameter called star square graceful deficiency number of a graph is defined and the star square graceful deficiency number of the cycle ð¶3 is determined. Two new definitions namely, odd square graceful labeling and even square graceful labeling of a graph are defined with example.
Keywords |
Square graceful graph, odd square graceful graph, even square graceful graph, Star square graceful deficiency number of a graph |
INTRODUCTION |
The graphs considered in this paper are finite, undirected and without loops or multiple edges. Let G = (V, E) be a graph with p vertices and q edges. Terms not defined here are used in the sense of Harary[2].For number theoretic terminology [1] is followed |
A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. If the domain of the mapping is the set of vertices (edges / both) then the labeling is called a vertex (edge / Total) labeling. There are several types of graph labeling and a detailed survey is found in [4]. |
Rosa [6] introduced �� -valuation of a graph and Golomb [5] called it as graceful labeling. Several authors worked on graceful labeling, odd graceful labeling, even graceful labeling, super graceful labeling and skolem –graceful labeling. |
Recently the concept of square graceful labeling was introduced by T.Tharmaraj and P.B.Sarasija inthe year 2014.They studied the square graceful labeling of various graphs in [7, 8]. |
The following definitions are necessary for the present study. |
1.1 Definition |
The path on n vertices is denoted by���� . |
Definition [8] |
A complete bipartite graph��1,�� is called a star and it has n +1 vertices and n edges |
Definition |
The Corona ��1 ��2 of two graphs ��1 and ��2 is defined as the graph G by taking one copy of ��1(which has ��1 points) and ��1copies of ��2 and then joining the ith point of ��1 to every point in the ith copy of ��2. |
Definition |
Let the graphs ��1 and ��2 have disjoint vertex sets ��1 and ��2 and edge sets ��1 and ��2 respectively. Then their union G= ��1 ∪ ��2 is a graph with vertex set V= ��1 ∪ ��2 and edge set E= ��1 ∪ ��2.Clearly ��1 ∪ ��2 has ��1 + ��2 vertices and ��1 + ��2 edges. |
Definition |
The graph����@���� is obtained from ���� and m copies of ���� by identifying one pendant vertex of the ����â copy of ���� with ����â vertex of ���� where ���� is a path of length m-1. |
SQUARE GRACEFUL GRAPHS |
Definition[7] |
A ��. �� graph G = ��, �� is said to be a square graceful graph if there exists an injective function f: V �� → 0,1,2,3, … , ��2 such that the induced mapping ���� : E �� → 1,4,9, … , ��2 defined by ���� ���� = �� �� − �� �� is an injection. The function f is called a square graceful labeling of G. |
Example |
The square graceful labeling of the kite graph is given in figure a |
Figure a |
2.3 Observation |
The cycles ��3 and ��4 are not square graceful graphs |
CONCLUSION |
In this paper, the square graceful labeling of some graphs is studied. Examples of some non-square graceful graphs are observed. Star square graceful deficiency number of a graph is determined and the Star square graceful deficiency number of the cycle ��3 is determined. Odd square graceful labeling and even square graceful labeling are introduced. |
SCOPE FOR FURTHER STUDY |
The Star square graceful deficiency number of the cycle ����where n> 3, the wheel ���� , where n> 3,Odd square graceful labeling and even square graceful labeling of various graphs maybe studied. |
ACKNOWLEDGEMENTS |
The author is thankful to the anonymous Reviewer for the valuable comments and suggestions |
References |
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