On total edge irregularity strength of strong product of two cycles

Abstract

An edge irregular total k-labeling of a graph G = (V, E) is a labeling ℓ: V ∪ E → {1, 2,⋯, k} such that the total edge-weights wt(ab) = ℓ(a) + ℓ(ab) + ℓ(b) are different for all pairs of distinct edges. The minimum k for which the graph G has an edge irregular total Relabeling is called the total edge irregularity strength of G. The strong product G₁ G₂ of graphs G₁ and G₂ is the graph with V(G₁) × V(G₂) as the vertex set, and two distinct vertices (a₁,a₂) and (b₁,b₂) are adjacent whenever for each i ϵ {1, 2} either a_i = b_i or a_ib_i ϵ E(G_i). In this paper, we determine the exact value of the total edge irregularity strength of the strong product of two cycles C_n and C_m.