Total Edge Irregularity Strength of Strong Product of Cycles and Paths

An irregular assignment is a k -labeling of the edges : {1, 2, , } f E k → ... such that the vertex weights (label sums of edges incident with the vertex) are different for all vertices of G . The smallest k for which there is an irregular assignment is the irregularity strength. The notion of irregularity strength was introduced by Chartrand et al. [8] and studied by numerous authors, see [6,10,11,14,17,19,21]. Motivated by these papers, Ba c a et al. [5] started to investigate a total edge irregularity strength of graphs. For a graph G they define a labeling : {1,2, , } V E k φ ∪ → ... to be an edge irregular total k -labeling of the graph G if for every two different edges uv and u v ′ ′ of G one has ( ) = ( ) ( ) ( ) ( ) ( ) ( ) = ( ) w uv u uv v u u v v w u v φ φ φ φ φ φ φ φ ′ ′ ′ ′ ′ ′ + + ≠ + + . The minimum k for which the graph G has an edge irregular total k -labeling is called the total edge irregularity strength of G , ( ) tes G . In [5] is given a lower bound on the total edge irregularity strength of a graph, | ( ) | 2 ( ) 1 ( ) max , 3 2 E G G tes G ⎧ + Δ + ⎫ ⎡ ⎤ ⎡ ⎤ ≥ ⎨ ⎬ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎩ ⎭ , where ( ) G Δ is the maximum degree of