Modular total vertex irregularity strength of graphs

A (modular) vertex irregular total labeling of a graph $G$ of order $n$ is an assignment of positive integers from $1$ to $k$ to the vertices and edges of $G$ with the property that all vertex weights are distinct. The vertex weight of a vertex $v$ is defined as the sum of numbers assigned to the vertex $v$ itself and to the edge's incident, while the modular vertex weight is defined as the remainder of the division of the vertex weight by $n$. The (modular) total vertex irregularity strength of $G$ is the minimum $k$ for which such labeling exists. In this paper, we obtain estimations on the modular total vertex irregularity strength, and we evaluate the precise values of this invariant for certain graphs.