  The $k$-core percolation is a fundamental structural transition in complex
networks. Through the analysis of the size jump behaviors of $k$-core in the
evolution process of networks, we confirm that $k$-core percolation is
continuous phase transition when $k=1,2$ while it is a hybrid
first-order-second-order phase transition when $k\ge 3$. $2$-core percolation
belongs to different universality class from that of $1$-core (giant component)
percolation. The discontinuity of $k$-core percolation with $k\ge 3$ can be
concluded from largest size jump of $k$-core which will not disappear in the
thermodynamic limit while its continuous characteristic is reflected by second
largest size jump which converges to zero in power law as $N\to \infty$.
Furthermore, along with the previously known exponent $\beta=0.5$, we obtain a
set of exponents which are independent of $k$ when $k\ge 3$ and also different
from those critical exponents of $1$-core and $2$-core percolation.
