The b-chromatic number of powers of cycles

Abstract

A b-coloring of a graphGby k colors is a proper vertex coloring such that each color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k - 1 color classes. The b-chromatic number Xb(G) is the maximum integer k for which G has a b-coloring by k colors. Let Crn be the rth power of a cycle of order n. In 2003, Effantin and Kheddouci established the b-chromatic number Xb(Crn) for all values of n and r, except for 2r + 3 ≤ n ≤ 3r. For the missing cases they presented the lower bound L := min{n - r - 1, r + 1 + [ n-r-1/3]} and conjectured that Xb(C rn) = L. In this paper, we determine the exact value on Xb(Crn) for the missing cases. It turns out that Xb(Crn) > L for 2r + 3 ≤ n ≤ 2r + 3 + r-6/4. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.