### Totally irregular total labeling of some caterpillar graphs

#### Abstract

Assume that *G*(*V,E*) is a graph with *V* and *E* as its vertex and edge sets, respectively. We have *G* is simple, connected, and undirected. Given a function λ from a union of *V* and *E* into a set of *k*-integers from 1 until *k*. We call the function λ as a totally irregular total *k*-labeling if the set of weights of vertices and edges consists of different numbers. For any *u* ∈ *V*, we have a weight *wt*(*u*)=λ(*u*)+ ∑_{{uy ∈ E}} λ(*uy*). Also, it is defined a weight *wt*(*e*)= λ(*u*)+ λ(*uv*) + λ(*v*) for each *e*=*uv* ∈ *E*. A minimum *k* used in *k*-total labeling λ is named as a total irregularity strength of *G*, symbolized by ts(*G*). We discuss results on ts of some caterpillar graphs in this paper. The results are *ts*(S_{{p,2,2,q}}) = ⌈ (p+q-1)/2 ⌉ for *p*, *q* greater than or equal to 3, while ts(S_{{p,2,2,2,p}}) = ⌈(2p-1)/2 ⌉, *p* ≥ 4.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.2.5

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