Totally irregular total labeling of some caterpillar graphs

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 Vand 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(Sp,2,2,q) = ⌈p+q−1
2 ⌉ for p, q greater than or equal to 3, while ts(Sp,2,2,2,p) = ⌈2p−1 2 ⌉ , p ≥ 4