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We prove that ${\\rm sdepth}(J)\\geq n-\\nu_{o}(G)$ and ${\\rm sdepth}(S/J)\\geq n-\\nu_{o}(G)-1$, where $\\nu_{o}(G)$ is the ordered matching number of $G$. We also prove the inequalities ${\\rm sdepth}(J^k)\\geq {\\rm depth}(J^k)$ and ${\\rm sdepth}(S/J^k)\\geq {\\rm depth}(S/J^k)$, for every integer $k\\gg 0$, when $G$ is a bipartite graph. Moreover, we provide an elemen"},"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":"1604.00656","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-04-03T17:05:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5d3a5efce5fbbd78c8e8b042c11f7f5945ea0b05e433555543f6ce8fedbc681f","abstract_canon_sha256":"e83cf7489a0aa7fc725f415519be86ea84379aaef75c997ed0e0bd86883cbb2a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:48.762989Z","signature_b64":"pE8lmaCl3zIVdvb6vmRchpNwfldRvuzV+8NQgg9NSBsEtDjr4t0+R8PhcmY4GhyxZvgVN4TKrOmn/mfw2tdbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e842ff52a772b87ab9b95762d1978d947e9e48dccf9c162b594f5a7944bbdb67","last_reissued_at":"2026-05-18T01:17:48.762265Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:48.762265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Depth, Stanley depth and regularity of ideals associated to graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"S. A. Seyed Fakhari","submitted_at":"2016-04-03T17:05:07Z","abstract_excerpt":"Let $\\mathbb{K}$ be a field and $S=\\mathbb{K}[x_1,\\dots,x_n]$ be the polynomial ring in $n$ variables over $\\mathbb{K}$. Let $G$ be a graph with $n$ vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $J=J(G)$ is its cover ideal. We prove that ${\\rm sdepth}(J)\\geq n-\\nu_{o}(G)$ and ${\\rm sdepth}(S/J)\\geq n-\\nu_{o}(G)-1$, where $\\nu_{o}(G)$ is the ordered matching number of $G$. We also prove the inequalities ${\\rm sdepth}(J^k)\\geq {\\rm depth}(J^k)$ and ${\\rm sdepth}(S/J^k)\\geq {\\rm depth}(S/J^k)$, for every integer $k\\gg 0$, when $G$ is a bipartite graph. 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