{"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 $i$. 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.07482","kind":"arxiv","version":2},"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"}