The study of the b-choromatic number of some classes of fractal graphs. (4th July 2022) - Sattar, Tayyiba; Sardar, Muhammad Shoaib; Alaeiyan, Mehdi; Farahani, Moahmmad Reza; Cancan, Murat; Tas, Ziyattin
Abstract: In graph coloring, labels are assigned to graph elements according to certain constraints. Colors are a special case of graph labeling as well as in practical applications, graph coloring also poses some theoretical challenges. A topic related to graph coloring will be discussed in this study, i.e., b -chromatic number. In proper coloring, edges, vertices, or both of them are colored so that they are distinct from one another. A b -coloring of m colors of a graph G is similar to proper coloring in which at least one vertex from each color class is connected to ( m -1) other colors. The b-chromatic number of a graph G is the greatest positive number k such that G admits a b -coloring with k colors and is represented by ϕ ( G ). Fractals are geometric objects that are self-similar at multiple scales and their geometric measurements are different from fractal measurements. In this paper, we will evaluate the b -chromatic number of Fractal type graphs, i.e., Sierpinski network S( n ; Kk ) (where Kk is a complete graph of order k ) and Sierpinski gasket network S ( n ). Firstly, we will compute the b -chromatic number of S ( n ; K 3), S ( n ; K 4) and S ( n ; K 5) for n ≥ 2. After that, we will generalize the result for the Sierpinski network of complete graph Kk . In addition, we will also determine the b -choromatic number of Sierpinski gasket graph S ( n ). As an application, we will also determine the b -chromatic number of Sierpinski graph of house graph.