Computing the total H-irregularity strength of edge comb product of graphs

Abstract

A simple undirected graph = (V Γ, EΓ) admits an H-covering if every edge in E belongs to at least one subgraph of that is isomorphic to a graph H. For any graph admitting H-covering, a total labelling β : VΓ ∪EΓ→{1, 2, …, p} is called an H-irregular total p-labelling of Γ if every two different subgraphs H1 and H2 of isomorphic to H have distinct weights where the weight wβ(K) of subgraph K of Γ is defined as wf(K):=∑v∈VKf(v)+∑e∈EKf(e){w_f}\left( K \right): = \sum\limits_{v \in {V_K}} {f\left( v \right) + \sum\limits_{e \in {E_K}} {f\left( e \right)} } . The smallest number p for which a graph admits an H-irregular total p-labelling is called the total H-irregularity strength of Γ and is denoted by ths(Γ). In this paper, we determine the total H-irregularity strength of edge comb product of two graphs.

Keywords

• total h-irregular strength
• edge comb product graph
• cycle
• path
• primary 05c78, 05c76
• secondary 05c38