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In this note we prove that $G$ contains a heterochromatic cycle of length 4 if $G$ has $n\\geq 60$ vertices and $|CN(u)\\cup CN(v)|\\geq n-1$ for every pair of vertices $u$ and $v$ of $G$. This extends a result of Broersma et al. on the existence of heterochromatic cycles of length 3 or 4."},"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":"1201.3818","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-18T15:15:50Z","cross_cats_sorted":[],"title_canon_sha256":"c6023e03cfd97a01d6329a5ace2c8dc5174725fa79307f806d135c8206f82bca","abstract_canon_sha256":"3b258c2e4a572e8a30e3a97cf72a2d764b0d68efce7d621be9aa6820c024bc02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:17.914919Z","signature_b64":"93y89IO4zBoHB13NVe0bS1eJTgaBeIFKvUy9a5HLi3KrKG5ZT7b/F2G7QO84rTqgnSJVCk0NeAzVXGe9VrWoAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e1627f04bc3b9b5e41814c1e5567995c886ced31f56efdd36d1dd5d5b705f69","last_reissued_at":"2026-05-18T04:04:17.914201Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:17.914201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on heterochromatic cycles of length 4 in edge-colored graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Shenggui Zhang","submitted_at":"2012-01-18T15:15:50Z","abstract_excerpt":"Let $G$ be an edge-colored graph. A heterochromatic cycle of $G$ is one in which every two edges have different colors. For a vertex $v\\in V(G)$, let $CN(v)$ denote the set of colors which are assigned to the edges incident to $v$. In this note we prove that $G$ contains a heterochromatic cycle of length 4 if $G$ has $n\\geq 60$ vertices and $|CN(u)\\cup CN(v)|\\geq n-1$ for every pair of vertices $u$ and $v$ of $G$. 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