The k-core percolation on the Bethe lattice has been proposed as a simple
model of the jamming transition because of its hybrid first-order/second-order
nature. We investigate numerically k-core percolation on the four-dimensional
regular lattice. For k=4 the presence of a discontinuous transition is
clearly established but its nature is strictly first order. In particular, the
k-core density displays no singular behavior before the jump and its
correlation length remains finite. For k=3 the transition is continuous