The b-chromatic number of powers of cycles

A b-coloring of a graph G by 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 chi(b)(G) is the maximum integer k for which G has a b-coloring by k colors. Let C-n(r) be the rth power of a cycle of order n. In 2003, Effantin and Kheddouci established the b-chromatic number chi(b)(C-n(r)) 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 + left perpendicular n-r-1/3 right perpendicular} and conjectured that chi(b) (C-n(r)) - L. In this paper, we determine the exact value on chi(b)(C-n(r)) for the missing cases. It turns out that chi(b)(C-n(r)) > L for 2r + 3 <= n <= 2 r + 3 + r-6/4.