{"paper":{"title":"Birth of a giant $(k_1,k_2)$-core in the random digraph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Boris Pittel, Dan Poole","submitted_at":"2016-08-17T20:31:29Z","abstract_excerpt":"The $(k_1,k_2)$-core of a digraph is the largest sub-digraph with minimum in-degree and minimum out-degree at least $k_1$ and $k_2$ respectively. For $\\max\\{k_1, k_2\\} \\geq 2$, we establish existence of the threshold edge-density $c^*=c^*(k_1,k_2)$, such that the random digraph $D(n,m)$, on the vertex set $[n]$ with $m$ edges, asymptotically almost surely has a giant $(k_1,k_2)$-core if $m/n> c^*$, and has no $(k_1,k_2)$-core if $m/n<c^*$. Specifically, denoting $\\text{P}(\\text{Poisson}(z)\\ge k)$ by $p_k(z)$, we prove that $c^*=\\min\\limits_{z_1,z_2}\\max\\left\\{\\tfrac{z_1}{p_{k_1}(z_1)p_{k_2-1}("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05095","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}