DMGT

Discussiones Mathematicae Graph Theory 32(3) (2012) 427-434
DOI: https://doi.org/10.7151/dmgt.1619

Total Vertex Irregularity Strength of Disjoint Union of Helm Graphs

Ali Ahmad

College of Computer Science and Information Systems
Jazan University, Jazan
Kingdom of Saudi Arabia

E.T. Baskoro

Combinatorial Mathematics Research Group
Faculty of Mathematics and Natural Sciences
Institut Teknologi Bandung, Indonesia

M. Imran

Center for Advanced Mathematics and Physics (CAMP)
National University of Science and Technology (NUST)
H-12 Sector, Islamabad, Pakistan

Abstract

A total vertex irregular k-labeling φ of a graph G is a labeling of the vertices and edges of G with labels from the set {1,2, ..., k} in such a way that for any two different vertices x and y their weights wt(x) and wt(y) are distinct. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x. The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G. We have determined an exact value of the total vertex irregularity strength of disjoint union of Helm graphs.

Keywords: vertex irregular total k-labeling, Helm graphs, total vertex irregularity strength

2010 Mathematics Subject Classification: 05C78.

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Received 12 April 2011
Revised 20 July 2011
Accepted 25 July 2011