  Methods: Observing the large pc-scale Stokes I mm dust continuum emission
with the IRAM 30m telescope and the intermediate-scale (<0.1pc) polarized submm
dust emission with the Submillimeter Array toward a sample of 20 high-mass
star-forming regions allows us to quantify the dependence of the fragmentation
behaviour of these regions depending on the density and magnetic field
structures.
  Results: We infer density distributions n~r^{-p} of the regions with typical
power-law slopes p around ~1.5. There is no obvious correlation between the
power-law slopes of the density structures on larger clump scales (~1pc) and
the number of fragments on smaller core scales (<0.1pc). Comparing the
large-scale single-dish density profiles to those derived earlier from
interferometric observations at smaller spatial scales, we find that the
smaller-scale power-law slopes are steeper, typically around ~2.0. The
flattening toward larger scales is consistent with the star-forming regions
being embedded in larger cloud structures that do not decrease in density away
from a particular core. Regarding the magnetic field, for several regions it
appears aligned with filamentary structures leading toward the densest central
cores. Furthermore, we find different polarization structures with some regions
exhibiting central polarization holes whereas other regions show polarized
emission also toward the central peak positions. Nevertheless, the polarized
intensities are inversely related to the Stokes I intensities. We estimate
magnetic field strengths between ~0.2 and ~4.5mG, and we find no clear
correlation between magnetic field strength and the fragmentation level of the
regions. Comparison of the turbulent to magnetic energies shows that they are
of roughly equal importance in this sample. The mass-to-flux ratios range
between ~2 and ~7, consistent with collapsing star-forming regions.
