Sudden emergence of q-regular subgraphs in random graphs

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(3)-reg similar or equal to 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(3)-core similar or equal to 3.3509. For q > 3, the q-regular subgraph percolation threshold is found to coincide with that of the q-core.