  $k$-Core percolation has served as a paradigmatic model of discontinuous
percolation for a long time. Recently it was revealed that the order parameter
of $k$-core percolation of random networks additionally exhibits critical
behavior. Thus $k$-core percolation exhibits a hybrid phase transition. Unlike
the critical behaviors of ordinary percolation that are well understood, those
of hybrid percolation transitions have not been thoroughly understood yet.
Here, we investigate the critical behavior of $k$-core percolation of
Erd\H{o}s-R\'enyi networks. We find numerically that the fluctuations of the
order parameter and the mean avalanche size diverge in different ways. Thus, we
classify the critical exponents into two types: those associated with the order
parameter and those with finite avalanches. The conventional scaling relations
hold within each set, however, these two critical exponents are coupled.
Finally we discuss some universal features of the critical behaviors of
$k$-core percolation and the cascade failure model on multiplex networks.
