Sudden emergence of q-regular subgraphs in random graphs
arXiv:cond-mat/0603819 · doi:10.1209/epl/i2006-10070-4
Abstract
We investigate the computationally hard problem whether a random graph of finite average vertex degree has an extensively large -regular subgraph, i.e., a subgraph with all vertices having degree equal to . We reformulate this problem as a constraint-satisfaction problem, and solve it using the cavity method of statistical physics at zero temperature. For , we find that the first large -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 , the -regular subgraph percolation threshold is found to coincide with that of the -core.
7 pages, 5 figures