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Erd\\H{o}s and R\\'enyi (1960) conjectured that the limit\n  $$ \\lim_{n \\to \\infty} \\Pr\\{G(n,\\textstyle{n\\over 2}) is planar}} $$ exists and is a constant strictly between 0 and 1. \\L uczak, Pittel and Wierman (1994) proved this conjecture and Janson, \\L uczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability.\n  In this paper we determine the exact probability of a random graph being planar near the critical point $M=n/2$. For each $\\lambda$, we find an exact analytic expression for\n  $$ p(\\lambda)"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1204.3376","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-16T07:14:14Z","cross_cats_sorted":[],"title_canon_sha256":"07da1c1a1a9f2eb9b65c071832e3b729d12c56547c18823043a2077b036fb31f","abstract_canon_sha256":"de3a42c0a08f0bccecfc928296d96a51a6794a85b9eb701f52d3204a728878e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:44.698218Z","signature_b64":"qngLBZqJHng8NPyl9CbqBlaNZ+yy+yY/wgQVhjZZ9u5JjOaiKHcTAEiT8VUEdAQ+UvuYGFn8hQ2ZYRi0BhtTBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b3abce65a610e9397d2e95f557bf5028368a9fbac39a0bb318d36a6ce0a0c478","last_reissued_at":"2026-05-18T03:56:44.697529Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:44.697529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the probability of planarity of a random graph near the critical point","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Juanjo Ru\\'e, Marc Noy, Vlady Ravelomanana","submitted_at":"2012-04-16T07:14:14Z","abstract_excerpt":"Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\\H{o}s and R\\'enyi (1960) conjectured that the limit\n  $$ \\lim_{n \\to \\infty} \\Pr\\{G(n,\\textstyle{n\\over 2}) is planar}} $$ exists and is a constant strictly between 0 and 1. \\L uczak, Pittel and Wierman (1994) proved this conjecture and Janson, \\L uczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability.\n  In this paper we determine the exact probability of a random graph being planar near the critical point $M=n/2$. 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