{"paper":{"title":"$\\delta_k$-small sets in graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Asen Bojilov, Nedyalko Nenov","submitted_at":"2012-11-15T18:25:28Z","abstract_excerpt":"Let $G$ be a simple $n$-vertex graph and $W\\subseteq\\V(G)$. We say that $W$ is a $\\delta_k$-small set if $$ \\sqrt[k]{\\frac{\\sum_{v\\in W}d^k(v)}{\\abs W}}\\leq n-\\abs W. $$ Let $\\varphi^{(k)}(G)$ denote the smallest natural number $r$ such that $\\V(G)$ decomposes into $r$ $\\delta_k$-small sets, and let $\\alpha^{(k)}(G)$ denote the maximal number of vertices in a $\\delta_k$-small set of $G$. In this paper we obtain bounds for $\\alpha^{(k)}(G)$ and $\\varphi^{(k)}(G)$. Since $\\varphi^{(k)}(G)\\leq\\omega(G)\\leq\\chi(G)$ and $\\alpha(G)\\leq\\alpha^{(k)}(G)$, we obtain also bounds for the clique number $\\o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3689","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"}