{"paper":{"title":"Cohen-Macaulay, Shellable and unmixed clutters with a perfect matching of K\\\"onig type","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Enrique Reyes, Rafael H. Villarreal, Susan Morey","submitted_at":"2007-08-22T23:45:08Z","abstract_excerpt":"Let $\\mathcal{C}$ be a clutter with a perfect matching $e_1,...,e_g$ of K\\\"onig type and let $\\Delta_\\mathcal{C}$ be the Stanley-Reisner complex of the edge ideal of $\\mathcal{C}$. If all c-minors of $\\mathcal{C}$ have a free vertex and $\\mathcal{C}$ is unmixed, we show that $\\Delta_\\mathcal{C}$ is pure shellable. We are able to describe, in combinatorial and algebraic terms, when $\\Delta_\\mathcal{C}$ is pure. If $\\mathcal{C}$ has no cycles of length 3 or 4, then it is shown that $\\Delta_\\mathcal{C}$ is pure if and only if $\\Delta_\\mathcal{C}$ is pure shellable (in this case $e_i$ has a free v"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.3111","kind":"arxiv","version":3},"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"}