  The Fermi Large Area Telescope (LAT) revealed that blazars, representing the
most extreme radio-loud active galactic nuclei (AGN) population, dominate the
census of the gamma-ray sky, and a significant correlation was found between
radio and gamma-ray emission in the 0.1-100 GeV energy range. However, the
possible connection between radio and very high energy (VHE, E>0.1 TeV)
emission still remains elusive, owing to the lack of a homogeneous coverage of
the VHE sky. The main goal of this work is to quantify and assess the
significance of a possible connection between the radio emission on parsec
scale measured by the very long baseline interferometry (VLBI) and GeV-TeV
gamma-ray emission in blazars, which is a central issue for understanding the
blazar physics and the emission processes. We investigate the radio VLBI and
high energy gamma-ray emission by using two large and unbiased AGN samples
extracted from the first and second Fermi-LAT catalogs of hard gamma-ray
sources detected above 10 GeV (1FHL) and 50 GeV (2FHL). For comparison, we
perform the same correlation analysis by using the 0.1-300 GeV gamma-ray energy
flux provided by the third Fermi-LAT source catalog. We find that the
correlation strength and significance depend on the gamma-ray energy range with
a different behavior among the blazar sub-classes. Overall, the radio and
gamma-ray emission above 10 GeV turns out to be uncorrelated for the full
samples and for all of the blazar sub-classes with the exception of high
synchrotron peaked (HSP) objects, which show a strong and significant
correlation. On the contrary, when 0.1-300 GeV gamma-ray energies are
considered, a strong and significant correlation is found for the full blazar
sample as well as for all of the blazar sub-classes. We interpret and explain
this correlation behavior within the framework of the blazar spectral energy
distribution properties.
