  The so-called C2-DIBs are a class of very weak bands that fall in the blue
part of the optical spectrum and are associated with high column densities of
the C2 molecule. DIB profile structures constrain potential molecular carriers,
but their measurement requires high S/N and spectra and the use of sightlines
without Doppler splitting, as typical for a single-cloud situation. Spectra
from the ESO Diffuse Interstellar Bands Large Exploration Survey (EDIBLES)
conducted at the VLT (ESO/Paranal) were explored to identify single-cloud and
high C2 column sightlines, extract the corresponding C2-DIBs and study their
strengths and profiles, and to investigate in detail any sub-structures. The
target selection was made based on profile-fitting of the Sodium doublets and
the detection of C2 lines. The C2 (2-0) Phillips system was fitted using a
physical model of the host cloud. C2 column densities, temperatures as well as
gas densities were derived for each sightline. 18 known C2-DIBs and 8 strong
non-C2 DIBs were extracted towards 8 targets, comprising 7 single-cloud and one
multi-cloud line-of-sights. Correlational studies revealed a tight association
of the former group with the C2 columns, whereas the non-C2 DIBs are primarily
correlated with reddening. We report three new weak diffuse band candidates. We
show for the first time that at least 14 C2-DIBs exhibit spectral
sub-structures which are consistent with unresolved rotational branches of
molecular carriers. The variability of their peak separations among the bands
for a given sightline implies that their carriers are different molecules with
quite different sizes. We also illustrate how profiles of the same DIB vary
among targets and as a function of physical parameters and provide tables
defining the sub-structures to be compared with future models and experimental
results.
