On total vertex-irregular labellings for several types of trees

Abstract

For a simple graph G with the vertex set V(G) and the edge set E(G), a labelling λ : V(G)∪E(G) → {1,2,..., κ} is called a vertex-irregular total κ-labelling of G if for any two different vertices x and y in V(G), we have wt(x) ≠ wt(y) where wt(x) = λ(x) + Σ _xzεE(G) λ(xz). The to-tal vertex-irregular strength, denoted by tvs(G), is the smallest positive integer κ for which G has a vertex-irregular total κ-labelling. In this paper, we determine the total vertex-irregular strength for various types of trees, namely complete k-ary trees, a subdivision of stars, and a subdivision of particular type of caterpillars.