2-Distance Strong b-coloring of Perfect 𝜟-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.