bb-chromatic numbers of powers of paths and cycles

The bb-chromatic number χb(G)χb(G) of a graph GG is the maximum number kk for which there is a mapping f:V(G)→{1,2,…,k}f:V(G)→{1,2,…,k} such that f(x)≠f(y)f(x)≠f(y) for each edge xyxy and for each 1≤i≤k1≤i≤k there is a vertex xixi with f(xi)=if(xi)=i adjacent to some yijyij with f(yij)=jf(yij)=j for each j≠ij≠i. Effantin and Kheddouci (2003)  [8] gave the exact values for χb(Pnp) and χb(Cnp), except for the case 2p+3≤n≤3p2p+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⌋.