  The degree-restricted random process is a natural algorithmic model for
generating graphs with degree sequence D_n=(d_1, \ldots, d_n): starting with an
empty n-vertex graph, it sequentially adds new random edges so that the degree
of each vertex v_i remains at most d_i. Wormald conjectured in 1999 that, for
d-regular degree sequences D_n, the final graph of this process is similar to a
uniform random d-regular graph.
  In this paper we show that, for degree sequences D_n that are not nearly
regular, the final graph of the degree-restricted random process differs
substantially from a uniform random graph with degree sequence D_n. The
combinatorial proof technique is our main conceptual contribution: we adapt the
switching method to the degree-restricted process, demonstrating that this
enumeration technique can also be used to analyze stochastic processes (rather
than just uniform random models, as before).
