Sudden emergence of q-regular subgraphs in random graphs

Details Sudden emergence of q-regular subgraphs in random graphs Sudden emergence of q-regular subgraphs in random graphs

2006-03-30

arxiv.org/abs/cond-mat/0603819v1

Europhys. Lett. 75, 8 (2006)

Physics

Condensed Matter

Statistical Mechanics

7 pages, 5 figures

Scientific paper

10.1209/epl/i2006-10070-4

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.

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