  This paper is part of a series devoted to the investigation of a large sample
of brightest cluster galaxies (BCGs), their properties and the relationships
between these and the properties of the host clusters. In this paper, we
compare the stellar population properties derived from high signal-to-noise,
optical long-slit spectra with the GALEX ultraviolet (UV) colour measurements
for 36 nearby BCGs to understand the diversity in the most rapidly evolving
feature in old stellar systems, the UV-upturn. We investigate: (1) the possible
differences between the UV-upturn of BCGs and those of a control sample of
ordinary ellipticals in the same mass range, as well as possible correlations
between the UV-upturn and other general properties of the galaxies; (2)
possible correlations between the UV-upturn and the properties of the host
clusters; (3) recently proposed scenarios where helium-sedimentation in the
cluster centre can produce an enhanced UV-upturn. We find systematic
differences between the UV-colours of BCGs and ordinary ellipticals, but we do
not find correlations between these colours and the properties of the host
clusters. Furthermore, the observations do not support the predictions made by
the helium-sedimentation model as an enhancer of the UV-upturn.
