We use the largest complete sample of 64 galaxy clusters (HIghest X-ray FLUx Galaxy Cluster Sample) with available high-quality X-ray data from Chandra, and apply 16 cool-core diagnostics to them, some of them new. We also correlate optical properties of brightest cluster galaxies (BCGs) with X-ray properties. To segregate cool core and non-cool-core clusters, we find that central cooling time, t_cool, is the best parameter for low redshift clusters with high quality data, and that cuspiness is the best parameter for high redshift clusters. 72% of clusters in our sample have a cool core (t_cool < 7.7 h_{71}^{-1/2} Gyr) and 44% have strong cool cores (t_cool <1.0 h_{71}^{-1/2} Gyr). For the first time we show quantitatively that the discrepancy in classical and spectroscopic mass deposition rates can not be explained with a recent formation of the cool cores, demonstrating the need for a heating mechanism to explain the cooling flow problem. [Abridged]

What is a cool-core cluster? a detailed analysis of the cores of the X-ray flux-limitedHIFLUGCScluster sample