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).