  The celebrated Little-Parks effect in mesoscopic superconducting rings has
recently gained great attention due to its potential to probe half-quantum
vortices in spin-triplet superconductors. However, despite the large number of
works reporting anomalous Little-Parks measurements attributed to
unconventional superconductivity, the general signatures of spin-triplet
pairing in the Little-Parks effect have not yet been systematically
investigated. Here we use Ginzburg-Landau theory to study the Little-Parks
effect in a spin-triplet superconducting ring that supports half-quantum
vortices; we calculate the field-induced Little-Parks oscillations of both the
critical temperature itself and the residual resistance resulting from thermal
vortex tunneling below the critical temperature. We observe two separate
critical temperatures with a single-spin superconducting state in between and
find that, due to the existence of half-quantum vortices, each minimum in the
upper critical temperature splits into two minima for the lower critical
temperature. From a rigorous calculation of the residual resistance, we confirm
that these two minima in the lower critical temperature translate into two
maxima in the residual resistance below and establish the general conditions
under which the two maxima can be practically resolved. In particular, we
identify a fundamental trade-off between sharpening each maximum and keeping
the overall magnitude of the resistance large. Our results will guide
experimental efforts in designing mesoscopic ring geometries for probing
half-quantum vortices in spin-triplet candidate materials on the device scale.
