On total edge irregularity strength of categorical product of cycle and path

Abstract

We investigate a modification of well known irregularity strength of graph, namely the total edge irregularity strength. An edge irregular total k-labeling ϕ: V ≼ E → {1, 2, …, k} of a graph G is a labeling of vertices and edges of G in such a way that for any two different edges uv and u'v' their weights ϕ(u) + ϕ(uv) + ϕ(v) and ϕ(u') + ϕ(u'v') + ϕ(v') are distinct. The total edge irregularity strength, tes(G), is defined as the minimum k for which G has an edge irregular total k-labeling.The main purpose of this paper is to solve the open problem posed by Ahmad and Bača.