The b-chromatic number χb(G) of a graph G is the maximum number k for which there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) for each edge xy and for each 1≤i≤k there is a vertex xi with f(xi)=i adjacent to some yij with f(yij)=j for each j≠i. Effantin and Kheddouci (2003) [8] gave the exact values for χb(Pnp) and χb(Cnp), except for the case 2p+3≤n≤3p they only proved that χb(Cnp)≥min{n−p−1,⌊n+2p+23⌋}. They then conjectured that this lower bound is in fact the exact value. In this paper, we confirm the conjecture for ⌊9p+104⌋≤n≤3p and disprove the conjecture for 2p+3≤n≤⌊9p+64⌋.