  We study the edge deletion process of random graphs near a k-core percolation
point. We find that the time-dependent number of edges in the process exhibits
critically divergent fluctuations. We first show theoretically that the k-core
percolation point is exactly given as the saddle-node bifurcation point in a
dynamical system. We then determine all the exponents for the divergence based
on a universal description of fluctuations near the saddle-node bifurcation.
