A new generalization of edge-irregular evaluations

Consider a simple graph $ G = (V, E) $ of size $ m $ with the vertex set $ V $ and the edge set $ E $. A modular edge-irregular total $ k $-labeling of a graph $ G $ is a labeling scheme for the vertices and edges with the labels $ 1, 2, \dots, k $ that allows the modular weights of any two different edges to be distinct, where the modular weight of an edge is the remainder of the division of the weight (i.e., the sum of the label of the edge itself and the labels of its two end vertices) by $ m $. The maximal integer $ k $, minimized over all modular edge-irregular total $ k $-labelings of the graph $ G $ is called the modular total edge-irregularity strength. In the paper, we generalize the approach to edge-irregular evaluations, introduce the notion of the modular total edge-irregularity strength and obtain its boundary estimation. For certain families of graphs, we investigate the existence of modular edge-irregular total labelings and determine the precise values of the modular total edge-irregularity strength in order to prove the sharpness of the lower bound.