On b-Coloring Analysis of Graphs: An Application to Spatial-Temporal Graph Neural Networks for Multi-Step Time Series Forecasting of Soil Moisture and pH in Companion Farming

Let $G$ be a pair of two sets $(V,E)$ with vertex set $V$ and edge set $E$. A proper coloring of a graph $G$ is a vertex coloring of it such that no two adjacent vertices in $G$ have the same color. By $b-$Coloring, we define a coloring of the vertex of $G$ such that each color class has at least one vertex that adjacent with all other color classes. The $b-chromatic$ number of graph $G$, denoted by $\varphi(G),$ is the largest integer $k$ such that graph $G$ has $b-$Coloring with $k$ colors. In this paper, we will explore some new lemmas or theorems regarding to $\varphi(G)$. Furthermore, to see the robust application of $b-$Coloring of graph, at the end of this paper we will illustrate the implementation of $b-$Coloring on spatial temporal graph neural networks (STGNN) multi-step time series forecasting on soil moisture and $pH$ of companion farming.