{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:UWMUXKDIV7YONYIDLEYTSRFPR3","short_pith_number":"pith:UWMUXKDI","schema_version":"1.0","canonical_sha256":"a5994ba868aff0e6e10359313944af8ed7572571766ce73b60b315651f6de721","source":{"kind":"arxiv","id":"1704.00246","version":2},"attestation_state":"computed","paper":{"title":"Toughness and spanning trees in $K_4$-minor-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dong Ye, M. N. Ellingham, Songling Shan, Xiaoya Zha","submitted_at":"2017-04-02T03:02:20Z","abstract_excerpt":"For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the number of components of a graph $G$. We show that if $G$ is a connected $K_4$-minor-free graph and\n  $$\n  c(G-S) \\;\\le\\; \\sum_{v \\in S} (f(v)-1)\n  \\quad\\hbox{for all $S \\subseteq V(G)$ with $S \\ne \\emptyset$}\n  $$ then $G$ has a spanning $f$-tree. Consequently, if $G$ is a $\\frac{1}{k-1}$-tough $K_4$-minor-free graph, then $G$ has a spanning $k$-tree. These resu"},"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":"1704.00246","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-02T03:02:20Z","cross_cats_sorted":[],"title_canon_sha256":"992d43110a885fed78950f20a06aeae298b82ffe4c22d1749d8d2c3f62171f10","abstract_canon_sha256":"a8d453dc1ce5d5dbca4d3fc60e430f073ee7d7fb3949d8faf671b01219194c82"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:56.741082Z","signature_b64":"TX1+KKmVIehjyFNEdeOK88c53Xk4yz2eLCp51NloGArw0WJ7qFR9fMhtGXmo6GMZ7S6qsqNvolYwk3dRI3hlDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5994ba868aff0e6e10359313944af8ed7572571766ce73b60b315651f6de721","last_reissued_at":"2026-05-17T23:41:56.740431Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:56.740431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Toughness and spanning trees in $K_4$-minor-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dong Ye, M. N. Ellingham, Songling Shan, Xiaoya Zha","submitted_at":"2017-04-02T03:02:20Z","abstract_excerpt":"For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the number of components of a graph $G$. We show that if $G$ is a connected $K_4$-minor-free graph and\n  $$\n  c(G-S) \\;\\le\\; \\sum_{v \\in S} (f(v)-1)\n  \\quad\\hbox{for all $S \\subseteq V(G)$ with $S \\ne \\emptyset$}\n  $$ then $G$ has a spanning $f$-tree. Consequently, if $G$ is a $\\frac{1}{k-1}$-tough $K_4$-minor-free graph, then $G$ has a spanning $k$-tree. 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