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K.Murugan^{*}Assistant Professor, Department of Mathematics, The M.D.T. Hindu College, Tirunelveli, Tamilnadu, India |

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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 Ã¯Â¿Â½Ã¯Â¿Â½ -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 byÃ¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ . |

Definition [8] |

A complete bipartite graphÃ¯Â¿Â½Ã¯Â¿Â½1,Ã¯Â¿Â½Ã¯Â¿Â½ is called a star and it has n +1 vertices and n edges |

Definition |

The Corona Ã¯Â¿Â½Ã¯Â¿Â½1 Ã¢ÂÂ Ã¯Â¿Â½Ã¯Â¿Â½2 of two graphs Ã¯Â¿Â½Ã¯Â¿Â½1 and Ã¯Â¿Â½Ã¯Â¿Â½2 is defined as the graph G by taking one copy of Ã¯Â¿Â½Ã¯Â¿Â½1(which has Ã¯Â¿Â½Ã¯Â¿Â½1 points) and Ã¯Â¿Â½Ã¯Â¿Â½1copies of Ã¯Â¿Â½Ã¯Â¿Â½2 and then joining the ith point of Ã¯Â¿Â½Ã¯Â¿Â½1 to every point in the ith copy of Ã¯Â¿Â½Ã¯Â¿Â½2. |

## Definition |

Let the graphs Ã¯Â¿Â½Ã¯Â¿Â½1 and Ã¯Â¿Â½Ã¯Â¿Â½2 have disjoint vertex sets Ã¯Â¿Â½Ã¯Â¿Â½1 and Ã¯Â¿Â½Ã¯Â¿Â½2 and edge sets Ã¯Â¿Â½Ã¯Â¿Â½1 and Ã¯Â¿Â½Ã¯Â¿Â½2 respectively. Then their union G= Ã¯Â¿Â½Ã¯Â¿Â½1 ∪ Ã¯Â¿Â½Ã¯Â¿Â½2 is a graph with vertex set V= Ã¯Â¿Â½Ã¯Â¿Â½1 ∪ Ã¯Â¿Â½Ã¯Â¿Â½2 and edge set E= Ã¯Â¿Â½Ã¯Â¿Â½1 ∪ Ã¯Â¿Â½Ã¯Â¿Â½2.Clearly Ã¯Â¿Â½Ã¯Â¿Â½1 ∪ Ã¯Â¿Â½Ã¯Â¿Â½2 has Ã¯Â¿Â½Ã¯Â¿Â½1 + Ã¯Â¿Â½Ã¯Â¿Â½2 vertices and Ã¯Â¿Â½Ã¯Â¿Â½1 + Ã¯Â¿Â½Ã¯Â¿Â½2 edges. |

Definition |

The graphÃ¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½@Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ is obtained from Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ and m copies of Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ by identifying one pendant vertex of the Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¢ÂÂ copy of Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ with Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¢ÂÂ vertex of Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ where Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ is a path of length m-1. |

## SQUARE GRACEFUL GRAPHS |

## Definition[7] |

A Ã¯Â¿Â½Ã¯Â¿Â½. Ã¯Â¿Â½Ã¯Â¿Â½ graph G = Ã¯Â¿Â½Ã¯Â¿Â½, Ã¯Â¿Â½Ã¯Â¿Â½ is said to be a square graceful graph if there exists an injective function f: V Ã¯Â¿Â½Ã¯Â¿Â½ → 0,1,2,3, … , Ã¯Â¿Â½Ã¯Â¿Â½2 such that the induced mapping Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ : E Ã¯Â¿Â½Ã¯Â¿Â½ → 1,4,9, … , Ã¯Â¿Â½Ã¯Â¿Â½2 defined by Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ = Ã¯Â¿Â½Ã¯Â¿Â½ Ã¯Â¿Â½Ã¯Â¿Â½ − Ã¯Â¿Â½Ã¯Â¿Â½ Ã¯Â¿Â½Ã¯Â¿Â½ 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 Ã¯Â¿Â½Ã¯Â¿Â½3 and Ã¯Â¿Â½Ã¯Â¿Â½4 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 Ã¯Â¿Â½Ã¯Â¿Â½3 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 Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½where n> 3, the wheel Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ , 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, , Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½82− 96,2009 |