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This extends a result of Chung and Graham from 1983 who showed that there exist (non-planar) $n$-vertex graphs with $O(n \\log n)$ edges that contain all trees on $n$ vertices as subgraphs and a result from Gol'dberg and Livshits from 1968 who showed that there exists a universal tree for $n$-vertex trees on $n^{O(\\log(n))}$ vertices.\n  Furthermore, we determine the number of vertices needed in the worst case for a planar graph"},"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":"2409.01678","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-09-03T07:42:58Z","cross_cats_sorted":[],"title_canon_sha256":"99adf4bb436bc9e6136113c978f6c818ffbb070b5db8df99a4bb79ad0a729987","abstract_canon_sha256":"40f3caf2c0767abf182522b8cc5cf182cc2937216ab1e96508d69e4b1d225b97"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:02:32.640826Z","signature_b64":"q1Ths63/gG0Yc4WfpHqiFtiHVjsbbzOcAoLl+TrATphdmFoZvx4Hf6/TgDgReANfMDubqNAZ9n4Vte21skqbCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d087b44f3c436644f85c0a0a76f1740ef30acb3f80f527f6ae7bb7468929b905","last_reissued_at":"2026-07-05T09:02:32.640330Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:02:32.640330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Subgraph-universal planar graphs for trees","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandra Wesolek, Helena Bergold, Joachim Orthaber, Manfred Scheucher, Robert Lauff, Vesna Ir\\v{s}i\\v{c}","submitted_at":"2024-09-03T07:42:58Z","abstract_excerpt":"We show that there exists an outerplanar graph on $O(n^{c})$ vertices for $c = \\log_2(3+\\sqrt{10}) \\approx 2.623$ that contains every tree on $n$ vertices as a subgraph. 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