{"paper":{"title":"Lower bounds for independence and $k$-independence number of graphs using the concept of degenerate degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Manouchehr Zaker","submitted_at":"2015-07-26T12:32:34Z","abstract_excerpt":"Let $G$ be a graph and $v$ any vertex of $G$. We define the degenerate degree of $v$, denoted by $\\zeta(v)$ as $\\zeta(v)={\\max}_{H: v\\in H}~\\delta(H)$, where the maximum is taken over all subgraphs of $G$ containing the vertex $v$. We show that the degenerate degree sequence of any graph can be determined by an efficient algorithm. A $k$-independent set in $G$ is any set $S$ of vertices such that $\\Delta(G[S])\\leq k$. The largest cardinality of any $k$-independent set is denoted by $\\alpha_k(G)$. For $k\\in \\{1, 2, 3\\}$, we prove that $\\alpha_{k-1}(G)\\geq {\\sum}_{v\\in G} \\min \\{1, 1/(\\zeta(v)+("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07194","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"}