On Local Antimagic b-Coloring and Its Application for STGNN Time Series Forecasting on Horizontal Farming

Abstract

This article discusses a local antimagic coloring which is a combination between antimagic labeling and coloring. It is a new notion. We define a vertex weight of  as  where  is the set of edges incident to . The bijection  is said to be a local antimagic labeling if for any two adjacent vertices, their vertex weights must be distinct. Furthermore  a coloring of a graph is a proper coloring of the vertices of  such that in each color class there exists a vertex having neighbors in all other  color classes. If we assign color on each vertex by the vertex weight  such that it induces a graph coloring satisfying coloring property, then this concept falls into a local antimagic coloring of graph. A local antimagic chromatic number, denoted by , is the maximum number of colors chosen for any colorings generated by local antimagic coloring of . In this paper we initiate to explore some new lemmas or theorems regarding to . Furthermore, to see the robust application of local antimagic coloring, at the end of this paper we will analyse the implementation of local antimagic coloring on Graph Neural Networks (GNN) multi-step time series forecasting on for NPK (Nitrogen, Phosphorus, and Potassium) concentration of companion plantations.