On The Total Irregularity Strength of Regular Graphs

Rismawati Ramdani (Department of Mathematics, Faculty of Sciences and Technologies, Universitas Islam Negeri Sunan Gunung Djati, Jalan A.H. Nasution No. 105,)
A.N.M. Salman (Combinatorial Mathematics Research Group, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha No. 10, Bandung 40132, Indonesia)
Hilda Assiyatun (Combinatorial Mathematics Research Group, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha No. 10, Bandung 40132, Indonesia)

Journal of Mathematical and Fundamental Sciences
Vol. 47 No. 3 (2015)

Let 𝐺 = (𝑉, 𝐸) be a graph. A total labeling 𝑓: 𝑉 ∪ 𝐸 → {1, 2, ⋯ , 𝑘} iscalled a totally irregular total 𝑘-labeling of 𝐺 if every two distinct vertices 𝑥 and𝑦 in 𝑉 satisfy 𝑤𝑓(𝑥) ≠ 𝑤𝑓(𝑦) and every two distinct edges 𝑥1𝑥2 and 𝑦1𝑦2 in 𝐸satisfy 𝑤𝑓(𝑥1𝑥2) ≠ 𝑤𝑓(𝑦1𝑦2), where 𝑤𝑓(𝑥) = 𝑓(𝑥) + Σ𝑥𝑧∈𝐸(𝐺) 𝑓(𝑥𝑧) and𝑤𝑓(𝑥1𝑥2) = 𝑓(𝑥1) + 𝑓(𝑥1𝑥2) + 𝑓(𝑥2). The minimum 𝑘 for which a graph 𝐺 hasa totally irregular total 𝑘-labeling is called the total irregularity strength of 𝐺,denoted by 𝑡𝑠(𝐺). In this paper, we consider an upper bound on the totalirregularity strength of 𝑚 copies of a regular graph. Besides that, we give a dual labeling of a totally irregular total 𝑘-labeling of a regular graph and we consider the total irregularity strength of 𝑚 copies of a path on two vertices, 𝑚 copies of a cycle, and 𝑚 copies of a prism 𝐶𝑛 □ 𝑃2.