  We analyze sensitive, high-dynamic-range, observations to search for
extended, diffuse, radio emission in relaxed and cool-core galaxy clusters. We
performed deep 1.4 GHz Very Large Array observations, of A1068, A1413, A1650,
A1835, A2029, and complemented our dataset with archival observations of
Ophiuchus. We find that, in the central regions of A1835, A2029, and Ophiuchus,
the dominant radio galaxy is surrounded by diffuse low-brightness radio
emission that takes the form of a mini-halo. We detect no diffuse emission in
A1650, at a surface brightness level of the other mini-halos. We find low
significance indications of diffuse emission in A1068 and A1413, although to be
classified as mini-halos they would require further investigation, possibly
with data of higher signal-to-noise ratio. In the Appendix, we report on the
serendipitous detection of a giant radio galaxy with a total spatial extension
of ~1.6 Mpc.
