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.