A Note on Edge Weightings Inducing Proper Vertex Colorings

An edge k -weighting of a graph G = ( V , E ) is a function l : E → { 1 , ⋯ , k } . For v ∈ V we denote by S ( v ) the multiset of weights on edges incident to v . We say that a weighting l induces a vertex coloring via a function f if f ( S ( v ) ) ≠ f ( S ( u ) ) for all adjacent vertices u , v ∈ V . One corresponding coloring parameter is χ f ( G ) , the f -neighbor-distinguishing index. It is the smallest value k such that a k -weighting induces a vertex coloring of G via f . In literature, several functions f , e.g., sums and products, and different additional constraints for the colorings have been studied. Thereby a lot of related coloring parameters arise. In this note, we introduce a class of functions, so-called dispersing functions. We prove bounds for three classes of coloring parameters, which are induced by edge weightings via arbitrary dispersing functions.