  We investigate the computationally hard problem whether a random graph of
finite average vertex degree has an extensively large $q$-regular subgraph,
i.e., a subgraph with all vertices having degree equal to $q$. We reformulate
this problem as a constraint-satisfaction problem, and solve it using the
cavity method of statistical physics at zero temperature. For $q=3$, we find
that the first large $q$-regular subgraphs appear discontinuously at an average
vertex degree $c_\reg{3} \simeq 3.3546$ and contain immediately about 24% of
all vertices in the graph. This transition is extremely close to (but different
from) the well-known 3-core percolation point $c_\cor{3} \simeq 3.3509$. For
$q>3$, the $q$-regular subgraph percolation threshold is found to coincide with
that of the $q$-core.
