(untitled) (European Journal of Pure and Applied Mathematics, 2024)
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 φ(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 φ(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.