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

A simple undirected graph = (V _Γ, E_Γ) 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 : V_Γ ∪E_Γ→{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 w_β(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.