Mulyono, - On totally irregular total labeling of caterpillars having even number of internal vertices with degree three. AIP Conference Proceedings.

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Abstract

The concept of graph labeling had been developed rapidly. Many results have been found related to various types of labelings [5]. "A mapping from a set of elements (vertex, edge or both of them) of G(V,E) into a set of integers is named as labeling. When the domain is V ∪E, the mapping is mentioned as a total labeling" [13]. Let f be a total labeling. The weights of a vertex or an edge are as follows: wt(u) = f(u) + ∑uv∈E f(uv) and wt(xy) = f(x) + f(y) + f(xy). The concepts of edge (vertex) irregular total labelings were initiated by Bacaˇ et al. [4]: "a total k-labeling f : V ∪E → {1,2,..., k} is defined to be an edge irregular total k-labeling of G if every two different edges e1,e2 ∈ E have the weights wt(e1) = wt(e2) and to be a vertex irregular total k-labeling if for every two distinct vertices u and v have weights wt(u) = wt(v). The minimum number k for which G has an edge irregular total k-labeling is called the total edge irregularity strength of G, tes(G). Analogously, the total vertex irregularity strength of G is the minimum k for which G has a vertex irregular total k-labeling" [4]. Moreover, " f is a totally irregular total k-labeling of G if the weights of any two distinct vertices are distinct and any two different edges have different weights. The total irregularity strength of G, ts(G), is the minimum number k for which G has a totally irregular total k-labeling" [14].

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