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Random graphs with p=c/n, have a.a.s. linear tree-depth when c>1, the tree-depth is Theta (log n) when c=1 and Theta (loglog n) for c<1. The result for c>1 is derived from the computation of tree-width and provides a more direct proof of a conjecture by Gao on the linearity of tree-width recently proved by Lee, Lee and Oum. We also show that"},"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":"1104.2132","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-12T08:10:32Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"0ca0dc3ef56b26c05118868e3636e57fd0b630fed117b5af5efdc12898dc2f2f","abstract_canon_sha256":"9a7c6351ef4e3a19091bcf2b1a6ad777e5dfdcef07e7682dcd0b72de7b6260bb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:15.159130Z","signature_b64":"4BCbYbUKxW0oxePtjsv8WSygD0OHLiyC+XKHSOQk0ahggeKTA4UzdIcjS8iFQWHMUZO0mirFeuMW0+AbgYPeCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd4aa9635e80de4a47f25b990168c6c4bc5a110d8b749e09a2a52f0b3023a310","last_reissued_at":"2026-05-18T04:02:15.158647Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:15.158647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the tree-depth of Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Guillem Perarnau, Oriol Serra","submitted_at":"2011-04-12T08:10:32Z","abstract_excerpt":"The tree-depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree-depth of random graphs. For dense graphs, p>> 1/n, the tree-depth of a random graph G is a.a.s. td(G)=n-O(sqrt(n/p)). Random graphs with p=c/n, have a.a.s. linear tree-depth when c>1, the tree-depth is Theta (log n) when c=1 and Theta (loglog n) for c<1. The result for c>1 is derived from the computation of tree-width and provides a more direct proof of a conjecture by Gao on the linearity of tree-width recently proved by Lee, Lee and Oum. 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