Total irregularity strength of some cubic graphs

Abstract

Let G = (V;E) be a graph. A total labeling ψ : V ⋃ E → {1, 2, ....k} is called totally irregular total k-labeling of G if every two distinct vertices u and v in V (G) satisfy wt(u) ≠wt(v); and every two distinct edges u1u2 and v1v2 in E(G) satisfy wt(u1u2) ≠ wt(v1v2); where wt(u) = ψ (u) + ∑uv∊E(G) ψ(uv) and wt(u1u2) = ψ(u1) + ψ(u1u2) + ψ(u2): The minimum k for which a graph G has a totally irregular total k-labeling is called the total irregularity strength of G, denoted by ts(G): In this paper, we determine the exact value of the total irregularity strength of cubic graphs.