{"paper":{"title":"A note on the Brush Numbers of Mycielski Graphs, $\\mu(G)$","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Johan Kok, Sunny Joseph Kalayathankal, Susanth C","submitted_at":"2015-01-15T10:35:43Z","abstract_excerpt":"The concept of the brush number $b_r(G)$ was introduced for a simple connected undirected graph $G$. The concept will be applied to the Mycielskian graph $\\mu(G)$ of a simple connected graph $G$ to find $b_r(\\mu(G))$ in terms of an \\emph{optimal orientation} of $G$. We prove a surprisingly simple general result for simple connected graphs on $n \\geq 2$ vertices namely: $b_r(\\mu(G))= b_r(\\mu^{\\rightarrow}(G)) = 2\\sum\\limits_{i=1}^{n}d^+_{G^{\\rightarrow}_{b_r(G)}}(v_i).$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.03623","kind":"arxiv","version":1},"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"}