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We study $AR(n, kP_3 \\cup tP_2)$, where $kP_3 \\cup tP_2$ is the linear forest of $k$ disjoint paths on three vertices and a matching of size $t$. Recently, Jie and Jin [Discrete Appl. Math. 386 (2026) 30-57] determined this number for $k\\geq 2$, $t\\geq\\frac{k^2-3k+4}{2}$ and $n=2t+3k$. Here we solve the spanning case $n=3k+2t$ for all $k"},"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":"2509.25949","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-09-30T08:44:23Z","cross_cats_sorted":[],"title_canon_sha256":"b704fc754a1c284cfa853756d67f0e6618b27620fca2e50bfd3b24ed75315177","abstract_canon_sha256":"d1c9c25712fca27e08cf83f39463b4e492f3d51e60d71172ef30967557595206"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:34.343630Z","signature_b64":"x+UGYpxLBZH78DS1kWPDwyV/bSVx6OwsVV5e6LrS6tcoztRQxBJbIj1VxcRKD8fQU3VXZXVAqOhQrMfDjY3iAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29c5b354379e1c1599b54a2dac55103441b0352d7f75b4a19fbf2e5aabca2c3b","last_reissued_at":"2026-05-18T03:09:34.342759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:34.342759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Anti-Ramsey Numbers for Spanning Linear Forests of 3-Vertex Paths and Matchings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ali Ghalavand, Xueliang Li","submitted_at":"2025-09-30T08:44:23Z","abstract_excerpt":"A subgraph in an edge-colored graph is called rainbow if all its edges have distinct colors. For a graph $G$ and an integer $n$, the anti-Ramsey number $AR(n,G)$ is the maximum number of colors in an edge-coloring of $K_n$ that contains no rainbow copy of $G$. We study $AR(n, kP_3 \\cup tP_2)$, where $kP_3 \\cup tP_2$ is the linear forest of $k$ disjoint paths on three vertices and a matching of size $t$. Recently, Jie and Jin [Discrete Appl. Math. 386 (2026) 30-57] determined this number for $k\\geq 2$, $t\\geq\\frac{k^2-3k+4}{2}$ and $n=2t+3k$. 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