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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.","authors_text":"Nguyen Thu Hang, Thanh Vu, Tran Duc Dung","cross_cats":[],"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.","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-05T05:19:06Z","title":"Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.03369","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-07T12:57:52.019153Z","id":"912e134a-e61c-431f-afff-2f6ace0e4afb","model_set":{"reader":"grok-4.3"},"one_line_summary":"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.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_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.","strongest_claim":"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.","weakest_assumption":"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."}},"verdict_id":"912e134a-e61c-431f-afff-2f6ace0e4afb"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7c95d3c2397bdbce0aa7a797110ac33593303356d8621dbcab58dfc3c96f058e","target":"record","created_at":"2026-05-20T01:05:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9ca86d3b0cca3c946872f0ca4af299ac811453ac1b12eca074dec4ae3f537ba7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-05T05:19:06Z","title_canon_sha256":"f6b7c1b69e6045214e5ce53c56c4b723424c9e036cee91da4f2f913080d2f93e"},"schema_version":"1.0","source":{"id":"2605.03369","kind":"arxiv","version":2}},"canonical_sha256":"6ee6b1f9f1a76942a28cc85dcbec7f00902a698c32de73fc1c56820ad8fd90f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ee6b1f9f1a76942a28cc85dcbec7f00902a698c32de73fc1c56820ad8fd90f2","first_computed_at":"2026-05-20T01:05:15.280579Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T01:05:15.280579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VVxNjCgmy1lu3Vo5PdX2Ui5x5tF4UsWi5TFSaxoyIS83ZzzY2fG1tDBRR6rb+VhBjLwW5YAcnSJM8nF1RMyBCA==","signature_status":"signed_v1","signed_at":"2026-05-20T01:05:15.281473Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.03369","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c95d3c2397bdbce0aa7a797110ac33593303356d8621dbcab58dfc3c96f058e","sha256:c3ce8ab06cd64c08316a3d646c9da639fca7e8d856739639f63b9da4212133a4"],"state_sha256":"1c63a2852f9b05f679cc750ed241ff1ae5ace015e62617f85aad6b15d0495dd6"}