On The Total Irregularity Strength of Regular Graphs
DOI:
https://doi.org/10.5614/j.math.fund.sci.2015.47.3.6Keywords:
cycle, dual labeling, path, prism, regular graph, the total irregularity strength, totally irregular total k-labeling.Abstract
Let 𝐺 = (?, 𝐸) be a graph. A total labeling ?: ? 𝐸 ? {1, 2, , ?} iscalled a totally irregular total ?-labeling of 𝐺 if every two distinct vertices and
in ? satisfy ?() ?() and every two distinct edges 12 and 12 in 𝐸
satisfy ?(12) ?(12), where ?() = ?() + Σ?𝐸(𝐺) ?() and
?(12) = ?(1) + ?(12) + ?(2). The minimum ? for which a graph 𝐺 has
a totally irregular total ?-labeling is called the total irregularity strength of 𝐺,
denoted by (𝐺). In this paper, we consider an upper bound on the total
irregularity strength of ? copies of a regular graph. Besides that, we give a dual labeling of a totally irregular total ?-labeling of a regular graph and we consider the total irregularity strength of ? copies of a path on two vertices, ? copies of a cycle, and ? copies of a prism 𝐶? ?2.
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