Total Vertex Irregularity Strength of 1-Fault Tolerant Hamiltonian Graphs

Abstract

Let G(V, E) be a simple graph. For a labeling ${\partial\,:\,V\,\cup\,E\,\rightarrow\,\{1,\,2,\,3,...,k\}}$ the weight of a vertex x is defined as ${wt(x)\,=\,\partial\, (x)\,+\,\sum_{xy\in E} \partial\,(xy).}$ ${\partial}$ is called a vertex irregular total k-labeling if for every pair of distinct vertices x and y ${wt(x)\,\neq\,wt(y)}$. The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G and it is denoted by tvs(G). In this paper we determine the total vertex irregularity strength of 1-fault tolerant hamiltonian graphs ${CH(n),\,H(n)}$ and W(m).