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We prove a surprisingly simple general result for simple connected graphs on $n \\geq 2$ vertices namely: $b_r(\\mu(G))= b_r(\\mu^{\\rightarrow}(G)) = 2\\sum\\limits_{i=1}^{n}d^+_{G^{\\rightarrow}_{b_r(G)}}(v_i).$"},"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":"1501.03623","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2015-01-15T10:35:43Z","cross_cats_sorted":[],"title_canon_sha256":"a17495878d482926c5d3d0c66d70e9c9f96955b5a4a46b58eca7e685637d8eed","abstract_canon_sha256":"6af54f959c5edb414adb215275e9937e3662b9ddf553220a72aba46df0b840a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:18.129592Z","signature_b64":"EjovR97MC8Y5NJw7fPSwL1VpPk8l2XeRPpj3La3k2+0rJc5t17RfG8x+dUa6vV/uChDhRV7W9+8JIjhG5UUnCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c5ce83a76f6d018bdd7825073161fc4ce0d8d0ba118df70626bc0350a839537","last_reissued_at":"2026-05-18T02:29:18.128968Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:18.128968Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $G$. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1501.03623","created_at":"2026-05-18T02:29:18.129070+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.03623v1","created_at":"2026-05-18T02:29:18.129070+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.03623","created_at":"2026-05-18T02:29:18.129070+00:00"},{"alias_kind":"pith_short_12","alias_value":"HROOQOTW63IB","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"HROOQOTW63IBRPOX","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"HROOQOTW","created_at":"2026-05-18T12:29:25.134429+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HROOQOTW63IBRPOXQJIHGFQ7YT","json":"https://pith.science/pith/HROOQOTW63IBRPOXQJIHGFQ7YT.json","graph_json":"https://pith.science/api/pith-number/HROOQOTW63IBRPOXQJIHGFQ7YT/graph.json","events_json":"https://pith.science/api/pith-number/HROOQOTW63IBRPOXQJIHGFQ7YT/events.json","paper":"https://pith.science/paper/HROOQOTW"},"agent_actions":{"view_html":"https://pith.science/pith/HROOQOTW63IBRPOXQJIHGFQ7YT","download_json":"https://pith.science/pith/HROOQOTW63IBRPOXQJIHGFQ7YT.json","view_paper":"https://pith.science/paper/HROOQOTW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.03623&json=true","fetch_graph":"https://pith.science/api/pith-number/HROOQOTW63IBRPOXQJIHGFQ7YT/graph.json","fetch_events":"https://pith.science/api/pith-number/HROOQOTW63IBRPOXQJIHGFQ7YT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HROOQOTW63IBRPOXQJIHGFQ7YT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HROOQOTW63IBRPOXQJIHGFQ7YT/action/storage_attestation","attest_author":"https://pith.science/pith/HROOQOTW63IBRPOXQJIHGFQ7YT/action/author_attestation","sign_citation":"https://pith.science/pith/HROOQOTW63IBRPOXQJIHGFQ7YT/action/citation_signature","submit_replication":"https://pith.science/pith/HROOQOTW63IBRPOXQJIHGFQ7YT/action/replication_record"}},"created_at":"2026-05-18T02:29:18.129070+00:00","updated_at":"2026-05-18T02:29:18.129070+00:00"}