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For a set $S$ of $k$ vertices of $G$, let $\\kappa (S)$ denote the maximum number $\\ell$ of edge-disjoint trees $T_1,T_2,...,T_\\ell$ in $G$ such that $V(T_i)\\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\\leq i,j\\leq \\ell$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $\\kappa_k(G)$, of $G$ is defined by $\\kappa_k(G)=$min$\\{\\kappa(S)\\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $\\kappa_2(G)=\\kappa(G)$"},"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":"1012.5710","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-28T07:48:14Z","cross_cats_sorted":[],"title_canon_sha256":"16218fd900dd83765085f2a19f0902472824dfb6cc084d4bcf855b372a7f8795","abstract_canon_sha256":"9c1bfea33d430ef0099d70b76fbc086c76e85ce2006bbfed0ac3a4409c2b4ec4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:22.444954Z","signature_b64":"IgCjWpvpk1YZAt+Yc6V/WhVr/uYN8xPXjOihoeV8pf2Ibgk1Bqo5AxF22myEgO5rY7ebJZOMted9GL3T2JaUCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f106696b01d1d5a9a6e27a8bf85f927affafd4acb9f122bba5852ada3918c36a","last_reissued_at":"2026-05-18T04:32:22.444400Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:22.444400Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The generalized connectivity of complete bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shasha Li, Wei Li, Xueliang Li","submitted_at":"2010-12-28T07:48:14Z","abstract_excerpt":"Let $G$ be a nontrivial connected graph of order $n$, and $k$ an integer with $2\\leq k\\leq n$. For a set $S$ of $k$ vertices of $G$, let $\\kappa (S)$ denote the maximum number $\\ell$ of edge-disjoint trees $T_1,T_2,...,T_\\ell$ in $G$ such that $V(T_i)\\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\\leq i,j\\leq \\ell$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $\\kappa_k(G)$, of $G$ is defined by $\\kappa_k(G)=$min$\\{\\kappa(S)\\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. 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