The b-chromatic number of power graphs of paths and cycles - San Diego, Immanuel T. (Animo Repository, 2005)
Let G be a graph on the vertices n x , x ,..., x 1 2 . 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 ϕ(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 1 ≤ i ≤ k , there exists a vertex x of color i adjacent to a vertex of every color j, where 1 . This paper discussed the b-chromatic number of the ≤ j ≠ i ≤ k 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 Cn when n ∉[2 p + 3, 3p]and a bound for the b-chromatic number of p Cn when n ∈[2 p + 3, 3p].