{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6TPDMXUAX7LOZURRHA6CUNHPBD","short_pith_number":"pith:6TPDMXUA","canonical_record":{"source":{"id":"1507.02815","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-10T09:15:59Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3e871ba65f3bfdb83afcbc5f27a8448f82fdd17ddc4e7f9a6349d08217a00940","abstract_canon_sha256":"59f521e37bf146163bac597570c24a0c3de75de3b77ef8f49d074811f293f8a0"},"schema_version":"1.0"},"canonical_sha256":"f4de365e80bfd6ecd231383c2a34ef08d594f6fbc106a94a76c77765901d2c89","source":{"kind":"arxiv","id":"1507.02815","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.02815","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"arxiv_version","alias_value":"1507.02815v1","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.02815","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"pith_short_12","alias_value":"6TPDMXUAX7LO","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6TPDMXUAX7LOZURR","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6TPDMXUA","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6TPDMXUAX7LOZURRHA6CUNHPBD","target":"record","payload":{"canonical_record":{"source":{"id":"1507.02815","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-10T09:15:59Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3e871ba65f3bfdb83afcbc5f27a8448f82fdd17ddc4e7f9a6349d08217a00940","abstract_canon_sha256":"59f521e37bf146163bac597570c24a0c3de75de3b77ef8f49d074811f293f8a0"},"schema_version":"1.0"},"canonical_sha256":"f4de365e80bfd6ecd231383c2a34ef08d594f6fbc106a94a76c77765901d2c89","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:03.139041Z","signature_b64":"NO2e/dFIL1G6CY7pqmAm5WmrvipCN7o0AFAV7O9s6RP9z6grvH76ao1KON0vEVCztuOHp4rCIEDnoKZu9OJIAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4de365e80bfd6ecd231383c2a34ef08d594f6fbc106a94a76c77765901d2c89","last_reissued_at":"2026-05-18T01:37:03.138498Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:03.138498Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.02815","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:37:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OIjIYMJ9CSW689fo70rBw7vtxgssU+mY4t/DXN4/6oG/cuHSWLfGula5MhZoyrNhCzp6HlwcAJfK903W4MHeBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T07:21:12.075221Z"},"content_sha256":"287cecdfadf30b58c04169490c97757eb46d670306ba9cefeef845ab4be6477e","schema_version":"1.0","event_id":"sha256:287cecdfadf30b58c04169490c97757eb46d670306ba9cefeef845ab4be6477e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6TPDMXUAX7LOZURRHA6CUNHPBD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Maria Axenovich, Pascal Weiner, Torsten Ueckerdt","submitted_at":"2015-07-10T09:15:59Z","abstract_excerpt":"Recently, Borodin, Kostochka, and Yancey (On $1$-improper $2$-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least $7$ can be $2$-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least $6$ admits a vertex coloring in $2$ colors such that each monochromatic component is a path of length at most $14$. Moreover, we show a list version of this result. On the other hand, for each positive integer $t\\geq 3$, we construct a planar graph of girth $4$ such that in any "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02815","kind":"arxiv","version":1},"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":"2026-05-18T01:37:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FksDwsOSJqVFlk5Rv8lVLMNYOIgqQS8JwgECYB316cwc1lwQ50Hpko1Xb6MyZpyDUHAcakIcxR1QuF4Mjjo1DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T07:21:12.075796Z"},"content_sha256":"6d184dd1c2c233a582bedd7f3faae89db4a513b6ab7fc9e98e08892d91cb047b","schema_version":"1.0","event_id":"sha256:6d184dd1c2c233a582bedd7f3faae89db4a513b6ab7fc9e98e08892d91cb047b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6TPDMXUAX7LOZURRHA6CUNHPBD/bundle.json","state_url":"https://pith.science/pith/6TPDMXUAX7LOZURRHA6CUNHPBD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6TPDMXUAX7LOZURRHA6CUNHPBD/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":"2026-05-19T07:21:12Z","links":{"resolver":"https://pith.science/pith/6TPDMXUAX7LOZURRHA6CUNHPBD","bundle":"https://pith.science/pith/6TPDMXUAX7LOZURRHA6CUNHPBD/bundle.json","state":"https://pith.science/pith/6TPDMXUAX7LOZURRHA6CUNHPBD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6TPDMXUAX7LOZURRHA6CUNHPBD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6TPDMXUAX7LOZURRHA6CUNHPBD","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":"59f521e37bf146163bac597570c24a0c3de75de3b77ef8f49d074811f293f8a0","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-10T09:15:59Z","title_canon_sha256":"3e871ba65f3bfdb83afcbc5f27a8448f82fdd17ddc4e7f9a6349d08217a00940"},"schema_version":"1.0","source":{"id":"1507.02815","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.02815","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"arxiv_version","alias_value":"1507.02815v1","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.02815","created_at":"2026-05-18T01:37:03Z"},{"alias_kind":"pith_short_12","alias_value":"6TPDMXUAX7LO","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6TPDMXUAX7LOZURR","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6TPDMXUA","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:6d184dd1c2c233a582bedd7f3faae89db4a513b6ab7fc9e98e08892d91cb047b","target":"graph","created_at":"2026-05-18T01:37:03Z","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":"Recently, Borodin, Kostochka, and Yancey (On $1$-improper $2$-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least $7$ can be $2$-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least $6$ admits a vertex coloring in $2$ colors such that each monochromatic component is a path of length at most $14$. Moreover, we show a list version of this result. On the other hand, for each positive integer $t\\geq 3$, we construct a planar graph of girth $4$ such that in any ","authors_text":"Maria Axenovich, Pascal Weiner, Torsten Ueckerdt","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-10T09:15:59Z","title":"Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02815","kind":"arxiv","version":1},"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":"sha256:287cecdfadf30b58c04169490c97757eb46d670306ba9cefeef845ab4be6477e","target":"record","created_at":"2026-05-18T01:37:03Z","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":"59f521e37bf146163bac597570c24a0c3de75de3b77ef8f49d074811f293f8a0","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-10T09:15:59Z","title_canon_sha256":"3e871ba65f3bfdb83afcbc5f27a8448f82fdd17ddc4e7f9a6349d08217a00940"},"schema_version":"1.0","source":{"id":"1507.02815","kind":"arxiv","version":1}},"canonical_sha256":"f4de365e80bfd6ecd231383c2a34ef08d594f6fbc106a94a76c77765901d2c89","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f4de365e80bfd6ecd231383c2a34ef08d594f6fbc106a94a76c77765901d2c89","first_computed_at":"2026-05-18T01:37:03.138498Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:03.138498Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NO2e/dFIL1G6CY7pqmAm5WmrvipCN7o0AFAV7O9s6RP9z6grvH76ao1KON0vEVCztuOHp4rCIEDnoKZu9OJIAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:03.139041Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.02815","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:287cecdfadf30b58c04169490c97757eb46d670306ba9cefeef845ab4be6477e","sha256:6d184dd1c2c233a582bedd7f3faae89db4a513b6ab7fc9e98e08892d91cb047b"],"state_sha256":"8234ceb6434a3d2647f9933a53313f4411a3d15a311a720d296561f8b6e74d91"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vhntUc/VHTCIP+YYHo2SowOkfqoKHHrkZs5K5ZKm0ILGLLrSRWLMJ3SMyeQgjYeTdoxmg1QUexxGUwsZoJa9Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-19T07:21:12.078344Z","bundle_sha256":"b547056454c7ad0a55a60ea3c7d997730575e1df342fb9e334a8874754207a7d"}}