{"paper":{"title":"On the existence of vertex-disjoint subgraphs with high degree sum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nicolas Lichiardopol, Shuya Chiba","submitted_at":"2015-03-11T11:10:24Z","abstract_excerpt":"For a graph $G$, we denote by $\\sigma_{2}(G)$ the minimum degree sum of two non-adjacent vertices if $G$ is non-complete; otherwise, $\\sigma_{2}(G) = +\\infty$. In this paper, we prove the following two results: (i) If $s_{1}, s_{2} \\ge 2$ are integers and $G$ is a non-complete graph with $\\sigma_{2}(G) \\ge 2(s_{1} + s_{2} + 1) - 1$, then $G$ contains two vertex-disjoint subgraphs $H_{1}$ and $H_{2}$ such that each $H_{i}$ is a graph of order at least $s_{i}+1$ with $\\sigma_{2}(H_{i}) \\ge 2s_{i} - 1$. (ii) If $s_{1}, s_{2} \\ge 2$ are integers and $G$ is a triangle-free graph of order at least $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03272","kind":"arxiv","version":5},"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"}