# Vertex graceful labeling of some classes of graphs.

## Authors

• A. P. Santhakumaran Hindustan Institute of Technology and Science.
• P. Balaganesan Hindustan Institute of Technology and Science.

## Keywords:

Caterpillar, One vertex union graphs, Regular spider, Actinia graph, Vertex-graceful labeling, Strong vertex-graceful labeling

## Abstract

A connected graph G = (V, E) of order atleast two, with order p and size q is called vertex-graceful if there exists a bijection f : V → {1, 2, 3, ··· p} such that the induced function f * : E → {0, 1, 2, ··· q - 1} defined by f * (uv) = (f(u) + f(v))(mod q) is a bijection. The bijection f is called a vertex-graceful labeling of G. A subset S of the set of natural numbers N is called consecutive if S consists of consecutive integers. For any set X, a mapping f : XN is said to be consecutive if f(X) is consecutive. A vertex-graceful labeling f is said to be strong if the function ƒ₁ : EN defined by ƒ₁(e) = f(u)+ f(v) for all edges e = uv in E forms a consecutive set. It is proved that one vertex union of odd number of copies of isomorphic caterpillars is vertex-graceful and any caterpillar is strong vertex-graceful. It is proved that a spider with even number of legs (paths) of equal length appended to each vertex of an odd cycle is vertex-graceful. It is also proved that the graph lA(mj , n) is vertex-graceful for both n and l odd, 0 ≤ in - 1, 1 ≤ jmi. Further, it is proved that the graph A(mj , n) is strong vertex-graceful for n odd, 0 ≤ i n - 1, 1 ≤ jmi.

## Author Biographies

### A. P. Santhakumaran, Hindustan Institute of Technology and Science.

Department of Mathematics.

### P. Balaganesan, Hindustan Institute of Technology and Science.

Research Scholar, Department of Mathematics.

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2018-03-15

## How to Cite

[1]
A. P. Santhakumaran and P. Balaganesan, “Vertex graceful labeling of some classes of graphs.”, Proyecciones (Antofagasta, On line), vol. 37, no. 1, pp. 19-43, Mar. 2018.

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