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.