Degree Equitable Chromatic Number of a Graph

Let \( G = (V, E) \) be a connected graph. A subset \( S \) of \( V \) is called a degree equitable set if the degrees of any two vertices in \( S \) differ by at most one. The minimum order of a partition of \( V \) into independent degree equitable sets is called the \emph{degree equitable chromatic number} of \( G \) and is denoted by \( \chi_{de}(G) \). In this paper, we initiate a study of this new coloring parameter.