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We prove that if $G$ is a connected triangle-free planar graph with $n$ vertices and $m$ edges, then $\\alpha_2(G) \\geq \\frac{6n - m - 1}{5}$. By Euler's Formula, this implies $\\alpha_2(G) \\geq \\frac{4}{5}n$. We also prove that if $G$ is a triangle-free planar graph on $n$ vertices with at most $n_3$ vertices of degr"},"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":"1709.04036","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-12T19:34:29Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3043e268ba709ca011e56e1b174989aed9e86d9784498b1042f733c5cd3e4bed","abstract_canon_sha256":"8d78570684ba4958907136208bdf0bd892fa2f61000760eed5b80b2626298165"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:58.752379Z","signature_b64":"s4x2n93/d4xzt1U5GOM8lC+HdRSjkGMq1k3E44KC1UjKIA9zBHQNAVeaYWGmBvV6UfldDgytf15DwdC5z31PCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6692b62eea31dc5b33b5111ef7dc03ab586e37dd5bab3b2da951f9dae3286063","last_reissued_at":"2026-05-18T00:22:58.751926Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:58.751926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Induced 2-degenerate Subgraphs of Triangle-free Planar Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Tom Kelly, Zden\\v{e}k Dvo\\v{r}\\'ak","submitted_at":"2017-09-12T19:34:29Z","abstract_excerpt":"A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. 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