{"paper":{"title":"Betti numbers of subgraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Duc Ho, Huy Tai Ha","submitted_at":"2015-10-15T10:23:22Z","abstract_excerpt":"Let $G$ be a simple graph on $n$ vertices. Let $H$ be either the complete graph $K_m$ or the complete bipartite graph $K_{r,s}$ on a subset of the vertices in $G$. We show that $G$ contains $H$ as a subgraph if and only if $\\beta_{i,\\alpha}(H) \\le \\beta_{i,\\alpha}(G)$ for all $i \\ge 0$ and $\\alpha \\in \\mathbb{Z}^n$. In fact, it suffices to consider only the first syzygy module. In particular, we prove that $\\beta_{1,\\alpha}(H) \\le \\beta_{1,\\alpha}(G)$ for all $\\alpha \\in \\mathbb{Z}^n$ if and only if $G$ contains a subgraph that is isomorphic to either $H$ or a multipartite graph $K_{2,\\dots,2,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.04463","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"}