2-Distance Strong b-coloring of Perfect ????-ary Tree

A 2-distance ????-coloring is a 2-distance coloring in which every color class contains a vertex which has a neighbor in every other color class. A 2-distance strong ????-coloring (2????????- coloring) is a 2-distance coloring in which every color class contains a vertex ???? such that there is a vertex ???? in every other color class satisfying the condition that the distance between ???? and ???? is at most 2. The 2-distance ????-chromatic number ????2????(????) (2????-number) is the largest integer ???? such that ???? admits a 2-distance ????-coloring with ???? colors and the 2-distance strong bchromatic number ????2????????(????) (2????????-number) is the maximum k such that ???? admits a 2????????-coloring with ???? colors. A tree with a special vertex called the root is called a rooted tree. A perfect ????- ary tree, is a rooted tree in which all internal vertices are of degree ???? and all pendant vertices are at the same level. In this paper, the exact bound of the 2????????-number of perfect ????-ary tree are obtained.