On Total Vertex Irregularity Strength of Hypercubes

• Main Author: Anita Hadi, Dian
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/45933
• Institution: Institut Teknologi Bandung

The concept of total vertex irregularity labelling was introduced by Ba?ca et al. (2007). Let G = (V;E) be a simple graph (without loop and multiple edges). A function : V [ E ! f1; 2; :::; kg is called a vertex-irregular total k????labelling of G if for any two different vertices u and v in V satisfy wt(u) 6= wt(v), where wt(u) = (u)+ P uw2E (uw): The total vertex irregularity strength of G, denoted by tvs(G), is the smallest positive integer k for which G has a vertex irregular total k-labelling. Baca, et al. (2007) derived the lower and upper bounds for any r-regular graph G with p vertices and q edges as follows: p+r r+1 tvs(G) p ???? r + 1: In addition, Nurdin, et al. (2010) also give a conjecture for a connected graph G that is tvs(G) = max n+n +1 ; l +n+n+1 +2 m ; :::; l + P i= ni +1 mo , where ni is the number of vertices of degree i = ; +1; +2; :::;, where and are the minimum and the maximum degree of G, respectively. In this thesis, we determine the total vertex irregularity strength of hypercube graph with 2n vertices namely, tvs(Qn) = 2n+n n+1 for n 14. This result achieves the lower bound given by Ba?ca, et al. (2007) and also strengthens the conjecture by Nurdin, et al.(2010).