### The b-chromatic number of powers of cycles

*Anja Kohl*

#### Abstract

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 χ

_{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 χ_{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+⌊ n-r-1 / 3⌋} and conjectured that χ_{b}(C_{n}^{r})=L. In this paper, we determine the exact value on χ_{b}(C_{n}^{r}) for the missing cases. It turns out that χ_{b}(C_{n}^{r})>L for 2r+3≤n≤2r+3+r-6 / 4.Full Text: PDF PostScript