{"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 - 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.","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","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.","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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.03369/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T15:27:42.383413Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"95b7326298c423f9e0b62d7564d47ee2845469cc5ae6d92d2f7b56e02463e5c2"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}