Context :

Aims : We present a photometric study of the early-type dwarf galaxy population of the Centaurus cluster , aiming at investigating the galaxy luminosity function ( LF ) and galaxy scaling relations down to the regime of galaxies with M _ { V } \sim - 10 mag .

Methods : On deep VLT/FORS1 V - and I -band images of the central part of the cluster , we identify cluster dwarf-galaxy candidates using both morphological and surface brightness selection criteria .
Photometric and structural parameters of the candidates are derived from analysis of their surface brightness profiles .
Fundamental scaling relations , such as the colour–magnitude and the magnitude-surface brightness relation , are used to distinguish the cluster from the background .

Results : We find a flat LF with a slope of \alpha = -1.14 \pm 0.12 for M _ { V } > -14 mag , when fitting a power law to the completeness-corrected galaxy number counts .
Applying a Schechter function leads to a consistent result of \alpha \sim - 1.1 .
When plotting the central surface brightness of a Sérsic model vs. the galaxy magnitude , we find a continuous relation for magnitudes -20 < M _ { V } < -10 mag , with only the brightest core galaxies deviating from this relation , in agreement with previous studies of other clusters .
Within our Centaurus dwarf galaxy sample we identify three very compact objects .
We discuss whether they belong to the class of the so-called compact elliptical galaxies ( cEs ) .
In a size–luminosity diagram ( R _ { \mathrm { eff } } vs . M _ { V } ) of early-type galaxies from a range of environments , we observe that R _ { \mathrm { eff } } slowly decreases with decreasing luminosity for -21 < M _ { V } < -13 mag and decreases more rapidly at fainter magnitudes .
This trend continues to the ultra-faint Local Group dwarf galaxies ( M _ { V } \sim - 4 mag ) .

Conclusions : The continuous central surface brightness vs. absolute magnitude relation and the smooth relation in the size–luminosity diagram over a wide range of magnitudes are consistent with the interpretation of dwarf galaxies and more massive elliptical galaxies being one family of objects with gradually changing structural properties .
The most massive core galaxies and the rare cE galaxies are the only exceptions .