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Given graphs $G$ and $H$ with $|V(H)|=k$, the lexicographic product $G[H]$ is the graph obtained by substituting every vertex of $G$ by a copy of $H$, and every edge of $G$ by a copy of $K_{k,k}$. %Our main results are the following. We prove that for a sufficiently long path $P$, a nonrepetitive coloring of $P[K_k]$ needs at least $3k+\\lfloor k/2\\rfloor$ colors. If $k>2$ then we need exactly $2k+1$ colors to nonrepetitively color $P["},"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":"1210.5607","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-20T11:48:03Z","cross_cats_sorted":[],"title_canon_sha256":"fb2bcbb118a3e71bf9bb0c6aa150838c7ce53a98f7ae21cd3b847c482b213f9f","abstract_canon_sha256":"21b1c5a34778dff3c4aa7afa63d55aea84cccf6c25082a15333fd7ab9705e522"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:20.293053Z","signature_b64":"vPJNzDbVenLQETwttIRnwPku/NeY8TpUhop39FvtEkQXaydZxef9/4zSn1sX3iElJXumNkBJWFyWN3+pGn+cAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f7b02163bd37c970b69f01b63803cc98ee7d5a13d9b141190e5650a129437cb","last_reissued_at":"2026-05-18T03:13:20.292567Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:20.292567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nonrepetitive colorings of lexicographic product of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Keszegh, Bal\\'azs Patk\\'os, Xuding Zhu","submitted_at":"2012-10-20T11:48:03Z","abstract_excerpt":"A coloring $c$ of the vertices of a graph $G$ is nonrepetitive if there exists no path $v_1v_2\\ldots v_{2l}$ for which $c(v_i)=c(v_{l+i})$ for all $1\\le i\\le l$. Given graphs $G$ and $H$ with $|V(H)|=k$, the lexicographic product $G[H]$ is the graph obtained by substituting every vertex of $G$ by a copy of $H$, and every edge of $G$ by a copy of $K_{k,k}$. %Our main results are the following. We prove that for a sufficiently long path $P$, a nonrepetitive coloring of $P[K_k]$ needs at least $3k+\\lfloor k/2\\rfloor$ colors. 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