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We prove that if $B$ and $R$ are two $C_4$-free graphs on the same vertex set $V$ and $G(B,R)$ is the complete graph, then there exists an $B$-clique $X$, an $R$-clique $Y$ and a clique $Z$ in $B$ and $R$, such that $V=X\\cup Y\\cup Z$. Further, if $x\\in Z$ then $x$ is one of the vertices of some double $C_5$ in $G(B,R)$. In particular, if also $G(B,R)$ does not contains a double $C_5$, then $V$"},"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":"1511.08772","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-27T19:39:05Z","cross_cats_sorted":[],"title_canon_sha256":"3f12236aa2904805e570782edf4a8ecd318cea269e7e80b6c65f857563e2e813","abstract_canon_sha256":"71c1c3b765905e4bc9184b95ad10122e4a0cfbc137f64a5ba87709bce6695622"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:48.137731Z","signature_b64":"ER5oaC3wXtV34epunf886KG3LA9IaXEDxQg/R97pHkrIymWijPI2iGHRwyIKktk6VAkq2hi3fAofMPlRXTC0Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d88921dcf540b8a038b7e66d5d25c957a776ff20968ade2c04940c3ba862aea1","last_reissued_at":"2026-05-18T01:25:48.137233Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:48.137233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cliques in the union of $C_4$-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abeer Othman, Eli Berger","submitted_at":"2015-11-27T19:39:05Z","abstract_excerpt":"Let $B$ and $R$ be two simple graphs with vertex set $V$, and let $G(B,R)$ be the simple graph with vertex set $V$, in which two vertices are adjacent if they are adjacent in at least one of $B$ and $R$. We prove that if $B$ and $R$ are two $C_4$-free graphs on the same vertex set $V$ and $G(B,R)$ is the complete graph, then there exists an $B$-clique $X$, an $R$-clique $Y$ and a clique $Z$ in $B$ and $R$, such that $V=X\\cup Y\\cup Z$. Further, if $x\\in Z$ then $x$ is one of the vertices of some double $C_5$ in $G(B,R)$. 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