Total vertex irregularity strength of comb product of two cycles

UIN Sunan Gunung Djati Bandung, A.H. Nasution No 105 Bandung, Indonesia

^* Corresponding author: rismawatiramdani@gmail.com

Abstract

Let G = (V (G),E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f : V (G) ∪ E(G) → {1,2...,k}. The vertex weight v under the labeling f is denoted by W_f(v) and defined by W_f(v) = f(v) + Σ_uv∈E(G)f(uv). A total k-labeling of G is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of G, denoted by tvs(G), is the minimum k such that G has a vertex irregular total k-labeling. This labeling was introduced by Bača, Jendrol', Miller, and Ryan in 2007. Let G and H be two connected graphs. Let o be a vertex of H . The comb product between G and H, in the vertex o, denoted by G⊳o H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the i-th copy of H at the vertex o to the i-th vertex of G. In this paper, we determine the total vertex irregularity strength of comb product of C_n and C_m where m ∈ {1,2}.