  We give lower bounds for the tunnel number of knots and handlebody-knots. We
also give a lower bound for the cutting number, which is a "dual" notion to the
tunnel number in the handlebody-knot theory. We provide necessary conditions
for constituent handlebody-knots by using $G$-family of quandles colorings. The
above two evaluations are obtained as the corollaries. Furthermore, we
construct handlebody-knots with arbitrary tunnel number and cutting number.
