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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}("},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1608.05095","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-08-17T20:31:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d3756aabef17309f3fa11dfa874e3680b944f3a0a0e9686a8da373653926ff8c","abstract_canon_sha256":"2be403408d482c0a88b45314c6f45f7c9245e2741c75b1637bc3e6c92f9f1dce"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:31.917936Z","signature_b64":"LkGBSbzWDMK143J3jbZqmj1eYeuyij46zp0jz0+q4CuxyrAGaPaA6XqXWyIDstpCu7pBOErQkU9vYee5yVoqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f82fd8f3ad65ba19255b94bb366f6bff9abddcebff86a71df16d15f2c8f14f2","last_reissued_at":"2026-05-18T01:08:31.917348Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:31.917348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $m/n<c^*$. 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