  Heat transport in the cuprate superconductors YBa$_2$Cu$_3$O$_{y}$ and
La$_{2-x}$Sr$_x$CuO$_4$ was measured at low temperatures as a function of
doping. A residual linear term kappa_{0}/T is observed throughout the
superconducting region and it decreases steadily as the Mott insulator is
approached from the overdoped regime. The low-energy quasiparticle gap
extracted from kappa_{0}/T is seen to scale closely with the pseudogap. The
ubiquitous presence of nodes and the tracking of the pseudogap shows that the
overall gap remains of the pure d-wave form throughout the phase diagram, which
excludes the possibility of a complex component (ix) appearing at a putative
quantum phase transition and argues against a non-superconducting origin to the
pseudogap. A comparison with superfluid density measurements reveals that the
quasiparticle effective charge is weakly dependent on doping and close to
unity.
