  We study both numerically and analytically what happens to a random graph of
average connectivity "alpha" when its leaves and their neighbors are removed
iteratively up to the point when no leaf remains. The remnant is made of
isolated vertices plus an induced subgraph we call the "core". In the
thermodynamic limit of an infinite random graph, we compute analytically the
dynamics of leaf removal, the number of isolated vertices and the number of
vertices and edges in the core. We show that a second order phase transition
occurs at "alpha = e = 2.718...": below the transition, the core is small but
above the transition, it occupies a finite fraction of the initial graph. The
finite size scaling properties are then studied numerically in detail in the
critical region, and we propose a consistent set of critical exponents, which
does not coincide with the set of standard percolation exponents for this
model. We clarify several aspects in combinatorial optimization and spectral
properties of the adjacency matrix of random graphs.
  Key words: random graphs, leaf removal, core percolation, critical exponents,
combinatorial optimization, finite size scaling, Monte-Carlo.
