Sudden emergence of q-regular subgraphs in random graphsM. Pretti1 and M. Weigt2
1 INFM-CNR, Dipartimento di Fisica, Politecnico di Torino Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
2 Institute for Scientific Interchange - Viale Settimio Severo 65, I-10133 Torino, Italy
received 31 March 2006; accepted 5 May 2006
published online 31 May 2006
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 and contain immediately about of all vertices in the graph. This transition is extremely close to (but different from) the well-known 3-core percolation point . For q>3, the q-regular subgraph percolation threshold is found to coincide with that of the q-core.
05.20.-y - Classical statistical mechanics.
02.50.-r - Probability theory, stochastic processes, and statistics.
89.20.-a - Interdisciplinary applications of physics.
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