{"paper":{"title":"Depth stability of edge ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"J\\\"urgen Herzog, Takayuki Hibi","submitted_at":"2016-02-18T17:43:37Z","abstract_excerpt":"Let $G$ be a connected finite simple graph and let $I_G$ be the edge ideal of $G$. The smallest number $k$ for which $\\depth S/I_G^k$ stabilizes is denoted by $\\dstab(I_G)$. We show that $\\dstab(I_G)<\\ell(I_G)$ where $\\ell(I_G)$ denotes the analytic spread of $I$. For trees we give a stronger upper bound for $\\dstab(I_G)$. We also show for any two integers $1\\leq a<b$ there exists a tree for which $\\dstab(I_G)=a$ and $\\ell(I_G)=b$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05890","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"}