The b-chromatic number of power graphs of paths and cycles
Abstract/Summary
Let G be a graph on the vertices n x x x ,..., , 2 1 . Its p-th power graph p G , has the same vertex set as G, with two distinct vertices of G adjacent in p G whenever there is a path of length at most p between them in G. The b-chromatic number of G, denoted by ) (G , is defined as the maximum number k of colors that can be used to color the vertices of G, such that no two adjacent vertices have the same color and for each color i, with k i 1 , there exists a vertex x of color i adjacent to a vertex of every color j, where k i j 1 . This paper discussed the b-chromatic number of the power graphs of paths and cycles. This paper is an exposition of the article entitled The b-chromatic number of some power graph by Brice Effantin and Hemamache Kheddouci which appeared in Discrete Mathematics and Theoretical Computer Science Volume 6 in 2003. This study gives the exact value for the b-chromatic number of power graphs of path. It also give the exact value for the b-chromatic number of p n C when ] 3 , 3 2 [ p p n + and a bound for the b-chromatic number of p n C when ] 3 , 3 2 [ p p n + .
