{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:CSITP54LTMZEAOM76NZQNPIHJL","short_pith_number":"pith:CSITP54L","canonical_record":{"source":{"id":"1305.6195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-27T12:32:29Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"fb42cd8062caf719ebf68de293c8028500783e5662262308954222d2a8ad2910","abstract_canon_sha256":"350aca2152cceefaf94f7e65b6a0cee814cb6ddf5cc6a674591070ef2273eb3f"},"schema_version":"1.0"},"canonical_sha256":"149137f78b9b3240399ff37306bd074ad3ff5bd90d70d8873cbd03c7a646b7a5","source":{"kind":"arxiv","id":"1305.6195","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.6195","created_at":"2026-05-18T03:11:23Z"},{"alias_kind":"arxiv_version","alias_value":"1305.6195v2","created_at":"2026-05-18T03:11:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6195","created_at":"2026-05-18T03:11:23Z"},{"alias_kind":"pith_short_12","alias_value":"CSITP54LTMZE","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CSITP54LTMZEAOM7","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CSITP54L","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:CSITP54LTMZEAOM76NZQNPIHJL","target":"record","payload":{"canonical_record":{"source":{"id":"1305.6195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-27T12:32:29Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"fb42cd8062caf719ebf68de293c8028500783e5662262308954222d2a8ad2910","abstract_canon_sha256":"350aca2152cceefaf94f7e65b6a0cee814cb6ddf5cc6a674591070ef2273eb3f"},"schema_version":"1.0"},"canonical_sha256":"149137f78b9b3240399ff37306bd074ad3ff5bd90d70d8873cbd03c7a646b7a5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:23.841133Z","signature_b64":"Nvoyab+X61RC/mShpjm5JI4sz2IkrTXid+Pt+l9SnA7LKkKKWCUdH7tIIGa2UGgXRIQY+iDfI5koqHF3K1XmAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"149137f78b9b3240399ff37306bd074ad3ff5bd90d70d8873cbd03c7a646b7a5","last_reissued_at":"2026-05-18T03:11:23.840495Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:23.840495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.6195","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:11:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TpxOhQ/2lweyjkN7k0RnndEktqblSoMehFcnKY9er2ZafU5vf3qmXPCbwUXMGgnMI9qbhvK9axE03nwpNbDwDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T15:41:00.373595Z"},"content_sha256":"5980ffe23e18246cb43d0a4fb81483685da91742d5d264b0a4ceed7c0716bc98","schema_version":"1.0","event_id":"sha256:5980ffe23e18246cb43d0a4fb81483685da91742d5d264b0a4ceed7c0716bc98"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:CSITP54LTMZEAOM76NZQNPIHJL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximum 4-degenerate subgraph of a planar graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"J\\'an Maz\\'ak, Robert Luko\\v{t}ka, Xuding Zhu","submitted_at":"2013-05-27T12:32:29Z","abstract_excerpt":"A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \\ge 2$ has a 4-degenerate induced subgraph containing at least $(38-d)/36$ of its vertices. This shows that every planar graph of order $n$ has a 4-degenerate induced subgraph of order more than $8/9 \\cdot n$. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more ve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6195","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:11:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/oak+ibHCHeIxjT6yfb6RdEHTqd+wh8Qxt3q4f1uLwq9HrU15Oc3efWe2KlNkYI4cHAMR2p71J616LYO/dzoBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T15:41:00.374306Z"},"content_sha256":"4fe5fd37fbad49a6dd75c1b6c68bc9c47958b452cefe94d6139d6853d013a204","schema_version":"1.0","event_id":"sha256:4fe5fd37fbad49a6dd75c1b6c68bc9c47958b452cefe94d6139d6853d013a204"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CSITP54LTMZEAOM76NZQNPIHJL/bundle.json","state_url":"https://pith.science/pith/CSITP54LTMZEAOM76NZQNPIHJL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CSITP54LTMZEAOM76NZQNPIHJL/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T15:41:00Z","links":{"resolver":"https://pith.science/pith/CSITP54LTMZEAOM76NZQNPIHJL","bundle":"https://pith.science/pith/CSITP54LTMZEAOM76NZQNPIHJL/bundle.json","state":"https://pith.science/pith/CSITP54LTMZEAOM76NZQNPIHJL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CSITP54LTMZEAOM76NZQNPIHJL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:CSITP54LTMZEAOM76NZQNPIHJL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"350aca2152cceefaf94f7e65b6a0cee814cb6ddf5cc6a674591070ef2273eb3f","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-27T12:32:29Z","title_canon_sha256":"fb42cd8062caf719ebf68de293c8028500783e5662262308954222d2a8ad2910"},"schema_version":"1.0","source":{"id":"1305.6195","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.6195","created_at":"2026-05-18T03:11:23Z"},{"alias_kind":"arxiv_version","alias_value":"1305.6195v2","created_at":"2026-05-18T03:11:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6195","created_at":"2026-05-18T03:11:23Z"},{"alias_kind":"pith_short_12","alias_value":"CSITP54LTMZE","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CSITP54LTMZEAOM7","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CSITP54L","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:4fe5fd37fbad49a6dd75c1b6c68bc9c47958b452cefe94d6139d6853d013a204","target":"graph","created_at":"2026-05-18T03:11:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \\ge 2$ has a 4-degenerate induced subgraph containing at least $(38-d)/36$ of its vertices. This shows that every planar graph of order $n$ has a 4-degenerate induced subgraph of order more than $8/9 \\cdot n$. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more ve","authors_text":"J\\'an Maz\\'ak, Robert Luko\\v{t}ka, Xuding Zhu","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-27T12:32:29Z","title":"Maximum 4-degenerate subgraph of a planar graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6195","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5980ffe23e18246cb43d0a4fb81483685da91742d5d264b0a4ceed7c0716bc98","target":"record","created_at":"2026-05-18T03:11:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"350aca2152cceefaf94f7e65b6a0cee814cb6ddf5cc6a674591070ef2273eb3f","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-27T12:32:29Z","title_canon_sha256":"fb42cd8062caf719ebf68de293c8028500783e5662262308954222d2a8ad2910"},"schema_version":"1.0","source":{"id":"1305.6195","kind":"arxiv","version":2}},"canonical_sha256":"149137f78b9b3240399ff37306bd074ad3ff5bd90d70d8873cbd03c7a646b7a5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"149137f78b9b3240399ff37306bd074ad3ff5bd90d70d8873cbd03c7a646b7a5","first_computed_at":"2026-05-18T03:11:23.840495Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:11:23.840495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Nvoyab+X61RC/mShpjm5JI4sz2IkrTXid+Pt+l9SnA7LKkKKWCUdH7tIIGa2UGgXRIQY+iDfI5koqHF3K1XmAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:11:23.841133Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.6195","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5980ffe23e18246cb43d0a4fb81483685da91742d5d264b0a4ceed7c0716bc98","sha256:4fe5fd37fbad49a6dd75c1b6c68bc9c47958b452cefe94d6139d6853d013a204"],"state_sha256":"a9797502b5012a381f2e3afd3cf9ea7a8b12872e2153c89f1b39edb823f7659a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RafjodrOKNnyEzxdGd0OnA/7Ys5BusPvrozlD/cZS5FjRTaBioXM6Fq+nmLu163qvdbAmt3BS/o4oIoKxqqyCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T15:41:00.380164Z","bundle_sha256":"d429cd0991fd92c0fe620b8e6bdcbb6c163b6b62ddd0b26c3371ed358e8d197c"}}