{"paper":{"title":"A stronger result on fractional strong colourings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Andrew D. King","submitted_at":"2010-09-30T22:41:37Z","abstract_excerpt":"Aharoni, Berger and Ziv recently proved the fractional relaxation of the strong colouring conjecture. In this note we generalize their result as follows. Let $k\\geq 1$ and partition the vertices of a graph $G$ into sets $V_1,..., V_r$, such that for $1\\leq i \\leq r$ every vertex in $V_i$ has at most $\\max\\{k, |V_i|-k \\}$ neighbours outside $V_i$. Then there is a probability distribution on the stable sets of $G$ such that a stable set drawn from this distribution hits each vertex in $V_i$ with probability $1/|V_i|$, for $1\\leq i\\leq r$. We believe that this result will be useful as a tool in p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.0032","kind":"arxiv","version":4},"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"}