{"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. Moreover, we provide an elemen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00656","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}