On the total vertex irregularity strength of comb product of two cycles and two stars

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 ∪ E → {1,2,3,...,k}. The vertex weight v under the labeling f is denoted by w_f(v) and defined by w_f(v) = f(v) + \sum_{uv \in{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 labelings were introduced by Baca, 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, denoted by G \rhd_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 two cycles and two stars.