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Square Graceful Labeling of Some Graphs

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.

K.Murugan

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