  We present two a posteriori error estimators for the virtual element method
(VEM) based on global and local flux reconstruction in the spirit of [5]. The
proposed error estimators are reliable and efficient for the $h$-, $p$-, and
$hp$-versions of the VEM. This solves a partial limitation of our former
approach in [6], which was based on solving local nonhybridized mixed problems.
Differently from the finite element setting, the proof of the efficiency turns
out to be simpler, as the flux reconstruction in the VEM does not require the
existence of polynomial, stable, divergence right-inverse operators. Rather, we
only need to construct right-inverse operators in virtual element spaces,
exploiting only the implicit definition of virtual element functions. The
theoretical results are validated by some numerical experiments on a benchmark
problem.
