  We investigate the magnetic field at the surface of 48 red giants selected as
promising for detection of Stokes V Zeeman signatures in their spectral lines.
We use the spectropolarimeters Narval and ESPaDOnS to detect circular
polarization within the photospheric absorption lines of our targets and use
the least-squares deconvolution (LSD) method. We also measure the classical
S-index activity indicator, and the stellar radial velocity. To infer the
evolutionary status of our giants and to interpret our results, we use
state-of-the-art stellar evolutionary models with predictions of convective
turnover times. We unambiguously detect magnetic fields via Zeeman signatures
in 29 of the 48 red giants in our sample. Zeeman signatures are found in all
but one of the 24 red giants exhibiting signs of activity, as well as 6 out of
17 bright giant stars.The majority of the magnetically detected giants are
either in the first dredge up phase or at the beginning of core He burning,
i.e. phases when the convective turnover time is at a maximum: this corresponds
to a 'magnetic strip' for red giants in the Hertzsprung-Russell diagram. A
close study of the 16 giants with known rotational periods shows that the
measured magnetic field strength is tightly correlated with the rotational
properties, namely to the rotational period and to the Rossby number Ro. Our
results show that the magnetic fields of these giants are produced by a dynamo.
Four stars for which the magnetic field is measured to be outstandingly strong
with respect to that expected from the rotational period/magnetic field
relation or their evolutionary status are interpreted as being probable
descendants of magnetic Ap stars. In addition to the weak-field giant Pollux, 4
bright giants (Aldebaran, Alphard, Arcturus, eta Psc) are detected with
magnetic field strength at the sub-gauss level.
