{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:MMPMWWKSVVPZUVICLYZIDOEQ4S","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"594d15b218c264b2ec3064e5c0fc5fcc2a98bd55db57dfafa38fee8926aaf81c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-09-10T09:48:27Z","title_canon_sha256":"ada6292e723b1e04e7d92af60691678ed5f33ffc207ca53969482d719d46279d"},"schema_version":"1.0","source":{"id":"1709.03882","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.03882","created_at":"2026-05-18T00:35:29Z"},{"alias_kind":"arxiv_version","alias_value":"1709.03882v1","created_at":"2026-05-18T00:35:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.03882","created_at":"2026-05-18T00:35:29Z"},{"alias_kind":"pith_short_12","alias_value":"MMPMWWKSVVPZ","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"MMPMWWKSVVPZUVIC","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"MMPMWWKS","created_at":"2026-05-18T12:31:31Z"}],"graph_snapshots":[{"event_id":"sha256:8202a7ea13c077cbedddaff5167a407838e2bfa6fcdc61ae329118a18c6753bd","target":"graph","created_at":"2026-05-18T00:35:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $G$ be a graph with $n$ vertices and let $S=\\mathbb{K}[x_1,\\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\\mathbb{K}$. Assume that $J(G)$ is the cover ideal of $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove that the sequences $\\{{\\rm sdepth}(S/J(G)^{(k)})\\}_{k=1}^\\infty$ and $\\{{\\rm sdepth}(J(G)^{(k)})\\}_{k=1}^\\infty$ are non-increasing and hence convergent. Suppose that $\\nu_{o}(G)$ denotes the ordered matching number of $G$. We show that for every integer $k\\geq 2\\nu_{o}(G)-1$, the modules $J(G)^{(k)}$ and $S/J(G)^{(k)}$ satisfy the Stanley's inequality. W","authors_text":"S. A. Seyed Fakhari","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-09-10T09:48:27Z","title":"Depth and Stanley depth of symbolic powers of cover ideals of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.03882","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a7e92a479ff974281090a8d4c59e5b215129d1f336e498390656120af2ef6d75","target":"record","created_at":"2026-05-18T00:35:29Z","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":"594d15b218c264b2ec3064e5c0fc5fcc2a98bd55db57dfafa38fee8926aaf81c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-09-10T09:48:27Z","title_canon_sha256":"ada6292e723b1e04e7d92af60691678ed5f33ffc207ca53969482d719d46279d"},"schema_version":"1.0","source":{"id":"1709.03882","kind":"arxiv","version":1}},"canonical_sha256":"631ecb5952ad5f9a55025e3281b890e496a3074e2e5ac0074912ada71cb59f92","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"631ecb5952ad5f9a55025e3281b890e496a3074e2e5ac0074912ada71cb59f92","first_computed_at":"2026-05-18T00:35:29.120389Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:35:29.120389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TtoD5Pbl+M2gFR+2c6+xhzIE3Vx57x51GOmNiAtkTN2Na98KRiJMF6lCxTYJwVgrkZWilq4BoCHhgajipViGBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:35:29.121165Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.03882","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a7e92a479ff974281090a8d4c59e5b215129d1f336e498390656120af2ef6d75","sha256:8202a7ea13c077cbedddaff5167a407838e2bfa6fcdc61ae329118a18c6753bd"],"state_sha256":"24d526b0ed9e23828f1c5efba53082e9a5e436d1f317c64bbfe2604770afc647"}