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Square Graceful Labeling of Some Graphs

K.Murugan*
Assistant Professor, Department of Mathematics, The M.D.T. Hindu College, Tirunelveli, Tamilnadu, India
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Abstract

A 𝑝. 𝑞 graph G = 𝑉, 𝐸 is said to be a square graceful graph ifthere exists an injective function f: V 𝐺 → 0,1,2,3, … , 𝑞2 such that the induced mapping 𝑓𝑝 : E 𝐺 → 1,4,9, … , 𝑞2 defined by 𝑓𝑝 𝑢𝑣 = 𝑓 𝑢 − 𝑓 𝑣 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 𝑋1, 𝑋2, … , 𝑋𝑛 , the graphs 𝑃𝑛 −1 1,2, … 𝑛 ,m𝐾1,𝑛 ∪ 𝑠𝐾1,𝑡 , 𝐾1,𝑖 , 𝑛𝑖 =1 𝑃𝑛 ⨀𝐾1 − 𝑒,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 𝐶3 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 ����ℎ copy of ���� with ����ℎ 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
image
Figure a
2.3 Observation
The cycles ��3 and ��4 are not square graceful graphs
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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

1 M.Apostal, Introduction to Analytic Number Theory, Narosa Publishing House, Second edition,1991.

2 Frank Harary, Graph Theory, Narosa Publishing House, New Delhi,2001.

[3].S.W.Golomb,How to number a graph in Graph theory and Computing, R.C.Read, ed., Academic Press, Newyork, pp23-7,1972

4 Joseph A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, 15, #DS6.,2008

5 M.A.Perumal, S.Navaneethakrishnan, S.Arockiaraj and A.Nagarajan, Super graceful labeling for some special graphs, IJRRAS, Vol9, Issue3,2011

[6].A.Rosa,On certain valuations of the vertices of a graph, Theory of Graphs (International symposium, Rome, July1966),Gorden and Breach, N.Y and Dunod Paris,pp349-355,1967

[7] T.Tharmaraj and P.B.Sarasija, Square graceful graphs, International Journal of Mathematics and Soft Computing, Vol.4, No.1 pp129-137,2014

[8] T.Tharmaraj and P.B.Sarasija, Some square graceful graphs, International Journal of Mathematics and Soft Computing, Vol.5, No.1 pp119- 127,2015

9 R.Vasuki and A.Nagarajan, Some Results on Super Mean Graphs, International Journal of Mathematics and Combinatorics.Vol.3, , ����82− 96,2009