Introduction for SuperHyperGraph Labeling and MultiLabeling

A finite hypergraph generalizes the classical graph model by allowing hyperedges that can connect any nonempty subset of vertices. Building on this foundation, a finite SuperHyperGraph is obtained through iterative application of the powerset construction, thereby creating nested families of vertex and edge sets that capture multi-layered relationships. Graph labeling assigns numbers or symbols to vertices and/or edges of a graph under rules, modeling constraints, optimization, or communication. In this paper, we define and study the mathematical properties of Graph Labeling, HyperGraph Labeling, SuperHyperGraph Labeling, Graph MultiLabeling, HyperGraph MultiLabeling, and SuperHyperGraph MultiLabeling.