A structural approach to the graceful coloring of a subclass of trees

Let [Formula: see text] and G be a simple graph. A graceful m-coloring of G is a proper vertex coloring of G using the colors in M which leads to a proper edge coloring using [Formula: see text] colors such that the associated color of each edge is the absolute difference between their end vertices. The graceful chromatic number [Formula: see text]. We prove that [Formula: see text] , where T is a tree with [Formula: see text]. Furthermore, we categorize the trees into three types along with its characterization and the related coloring algorithm are presented in this study.