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

dmtcs:631 - Discrete Mathematics & Theoretical Computer Science, March 24, 2013, Vol. 15 no. 1 - https://doi.org/10.46298/dmtcs.631
The b-chromatic number of powers of cycles

Authors: Anja Kohl 1

• 1 Faculty of Informatics/Mathematics [Dresden]

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

Volume: Vol. 15 no. 1
Section: Graph Theory
Published on: March 24, 2013
Accepted on: June 9, 2015
Submitted on: May 16, 2011
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]