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 $c_{\text{reg}}3\simeq 3.3546$ 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 $c_{\text{cor}}3\simeq 3.3509$. For $q>3$, the $q$-regular subgraph percolation threshold is found to coincide with that of the $q$-core.