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In this article, we show that if the minimum degree of an $n$-vertex graph $G$ is at least $n/2+k/2-1$ when $n\\ge 2k+3$, then $G$ is $k$-knitted. The minimum degree is sharp. As a corollary, we obtain that $k$-contraction-critical graphs are $\\left\\lceil\\frac{k}{8}\\right\\rceil$-connected."},"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":"1811.07482","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-19T03:34:21Z","cross_cats_sorted":[],"title_canon_sha256":"d3d61eefcc50216127856b96063dabe90e9496a47bba1b1d9e1151631b234b77","abstract_canon_sha256":"c9e6bd16f4ea8063611c134800cc42c08641728adb3a32d118e18041ea34e77e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:50.489479Z","signature_b64":"Ju7fYRoOcQQfyi0PHjlf34tz7Hy0OTOtHW9MpGQVtzHl3H2eSUwJM+wY3ElaSKeSxNHmOtJt7m9e0EwM/KKMDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27d17e8000fcbc68ca3b81a3de21fc953b50e9d9d12d511e44e53c3ad4304849","last_reissued_at":"2026-05-17T23:43:50.488876Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:50.488876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimum degree condition for a graph to be knitted","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gexin Yu, Martin Rolek, Runrun Liu","submitted_at":"2018-11-19T03:34:21Z","abstract_excerpt":"For a positive integer $k$, a graph is $k$-knitted if for each $k$-subset $S$ of vertices, and every partition of $S$ into disjoint parts $S_1, \\ldots, S_t$ for some $t\\ge 1$, one can find disjoint connected subgraphs $C_1, \\ldots, C_t$ such that $C_i$ contains $S_i$ for each $i$. 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