On total H-irregularity strength of graphs

Abstract

Let G be a graph with vertex set V and edge set E. Total labeling f : V(G) ∪ E(G) → {1, 2, 3, …, l} called a total l-labeling of a graph G. The total l-labeling is a total H-irregular l-labeling of graph G if for H ⊆ G, the total H-weights wtf (H) = ∑ v∈V(H) f(v) + ∑ e∈E(H) f(e) are distinct. The irregularity strength s(G) of a graph G is known as the minimum k for which G has an irregular assignment using labels at most k. The total H–irregular a–labeling from the minimum where the graph G is called the total H–irregularity strength of G, is denoted by tHs(G). In this paper, we have obtained tHs from linegrid, buttrefly, hexagonal and diamond graphs. To obtain the tHs, we begin to study the total irregularity strength of graph G with subgraph H.