{"paper":{"title":"On bounds for some graph invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carlos E. Valencia, Isidoro Gitler","submitted_at":"2005-10-18T17:44:23Z","abstract_excerpt":"Let $G$ be a graph without isolated vertices and let $\\alpha(G)$ be its stability number and $\\tau(G)$ its covering number. The {\\it $\\alpha_{v}$-cover} number of a graph, denoted by $\\alpha_{v}(G)$, is the maximum natural number $m$ such that every vertex of $G$ belongs to a maximal independent set with at least $m$ vertices. In the first part of this paper we prove that $\\alpha(G)\\leq \\tau(G)[1+\\alpha(G)-\\alpha_{v}(G)]$. We also discuss some conjectures analogous to this theorem.\n  In the second part we give a lower bound for the number of edges of a graph $G$ as a function of the stability "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0510387","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"}