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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 $\\o"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1211.3689","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-15T18:25:28Z","cross_cats_sorted":[],"title_canon_sha256":"8591fd624a24a3b76a4aec31b13710634c5153b429d4b89f7aab52e8a5cf0128","abstract_canon_sha256":"716528e5a5de212fc6b8d0e583b73abbea3e83c09e1c2607114780d46cee07b6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:39.056261Z","signature_b64":"/KP99+tBBkV0fH0Cwmhs/bUPbQ2T+qlNztd1f+c/hgSSENnHD49sz3pf3copDtVIs4pAW3sBKJ+GeOJT3S5uBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11349971e2c6f7ea2d75aa188c823a601cd3ea4305a16134b57904808e4b94f4","last_reissued_at":"2026-05-18T03:40:39.055563Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:39.055563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\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 $\\varphi^{(k)}(G)$. 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