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

Abstract

A 2-distance 𝑏-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 𝑏-coloring (2𝑠𝑏- coloring) is a 2-distance coloring in which every color class contains a vertex 𝑢 such that there is a vertex 𝑣 in every other color class satisfying the condition that the distance between 𝑢 and 𝑣 is at most 2. The 2-distance 𝑏-chromatic number 𝜒2𝑏(𝐺) (2𝑏-number) is the largest integer 𝑘 such that 𝐺 admits a 2-distance 𝑏-coloring with 𝑘 colors and the 2-distance strong bchromatic number 𝜒2𝑠𝑏(𝐺) (2𝑠𝑏-number) is the maximum k such that 𝐺 admits a 2𝑠𝑏-coloring with 𝑘 colors. A tree with a special vertex called the root is called a rooted tree. A perfect 𝛥- ary tree, is a rooted tree in which all internal vertices are of degree 𝛥 and all pendant vertices are at the same level. In this paper, the exact bound of the 2𝑠𝑏-number of perfect 𝛥-ary tree are obtained.