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We introduce the notion of $t$-admissible subgraphs of $G$ and show how to use them to compute the depth of the $t$-th symbolic powers of the cover ideal of $G$. As an application, we prove that \\[ \\depth\\big(S/J(C_n)^{(t)}\\big) = n - 1 - \\left\\lfloor \\frac{tn}{2t+1} \\right\\rfloor \\] for all $t \\ge 2$ and $n \\ge 3$, where $S = K[x_1,\\ldots,x_n]$ and $J(C_n)$ is the cover ideal of the cycle on $n$ vertices."},"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":"2605.03369","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-05T05:19:06Z","cross_cats_sorted":[],"title_canon_sha256":"f6b7c1b69e6045214e5ce53c56c4b723424c9e036cee91da4f2f913080d2f93e","abstract_canon_sha256":"9ca86d3b0cca3c946872f0ca4af299ac811453ac1b12eca074dec4ae3f537ba7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:05:15.281473Z","signature_b64":"VVxNjCgmy1lu3Vo5PdX2Ui5x5tF4UsWi5TFSaxoyIS83ZzzY2fG1tDBRR6rb+VhBjLwW5YAcnSJM8nF1RMyBCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ee6b1f9f1a76942a28cc85dcbec7f00902a698c32de73fc1c56820ad8fd90f2","last_reissued_at":"2026-05-20T01:05:15.280579Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:05:15.280579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"The depth of the t-th symbolic power of the cover ideal of a cycle graph C_n equals n-1 minus floor of t n over 2t plus 1.","cross_cats":[],"primary_cat":"math.AC","authors_text":"Nguyen Thu Hang, Thanh Vu, Tran Duc Dung","submitted_at":"2026-05-05T05:19:06Z","abstract_excerpt":"Let $G$ be a simple graph. We introduce the notion of $t$-admissible subgraphs of $G$ and show how to use them to compute the depth of the $t$-th symbolic powers of the cover ideal of $G$. As an application, we prove that \\[ \\depth\\big(S/J(C_n)^{(t)}\\big) = n - 1 - \\left\\lfloor \\frac{tn}{2t+1} \\right\\rfloor \\] for all $t \\ge 2$ and $n \\ge 3$, where $S = K[x_1,\\ldots,x_n]$ and $J(C_n)$ is the cover ideal of the cycle on $n$ vertices."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that depth(S/J(C_n)^{(t)}) = n - 1 - floor(tn/(2t+1)) for all t ≥ 2 and n ≥ 3, where S = K[x1,...,xn] and J(C_n) is the cover ideal of the cycle on n vertices.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the newly defined t-admissible subgraphs correctly encode the depth information for the symbolic powers, allowing the reduction to the stated formula for cycles without hidden restrictions on the graph or the base field.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The depth of the t-th symbolic power of the cover ideal of the cycle graph C_n equals n-1 minus the floor of tn over 2t+1, for t at least 2 and n at least 3.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The depth of the t-th symbolic power of the cover ideal of a cycle graph C_n equals n-1 minus floor of t n over 2t plus 1.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"93bb9afd165cf5510f656399cd5c301093620e8ced691e6600b352e82d81a147"},"source":{"id":"2605.03369","kind":"arxiv","version":2},"verdict":{"id":"912e134a-e61c-431f-afff-2f6ace0e4afb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T12:57:52.019153Z","strongest_claim":"We prove that depth(S/J(C_n)^{(t)}) = n - 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