{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:ZN5POEXYXBBWLWBNGLFS7NBD4A","short_pith_number":"pith:ZN5POEXY","canonical_record":{"source":{"id":"1706.02335","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-07T18:50:14Z","cross_cats_sorted":[],"title_canon_sha256":"de398c818f43ead7f08e4577da99a3e7ea65923e6d3a80600156f5da68e0079b","abstract_canon_sha256":"b6fac02ec7458bd059eb5a7d89b8d7174943b7a3a94e7b3aa853e3fadc6fb3bb"},"schema_version":"1.0"},"canonical_sha256":"cb7af712f8b84365d82d32cb2fb423e02cdbbf4b4b09ea61234537726fed45a0","source":{"kind":"arxiv","id":"1706.02335","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.02335","created_at":"2026-05-18T00:42:45Z"},{"alias_kind":"arxiv_version","alias_value":"1706.02335v1","created_at":"2026-05-18T00:42:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.02335","created_at":"2026-05-18T00:42:45Z"},{"alias_kind":"pith_short_12","alias_value":"ZN5POEXYXBBW","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZN5POEXYXBBWLWBN","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZN5POEXY","created_at":"2026-05-18T12:31:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:ZN5POEXYXBBWLWBNGLFS7NBD4A","target":"record","payload":{"canonical_record":{"source":{"id":"1706.02335","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-07T18:50:14Z","cross_cats_sorted":[],"title_canon_sha256":"de398c818f43ead7f08e4577da99a3e7ea65923e6d3a80600156f5da68e0079b","abstract_canon_sha256":"b6fac02ec7458bd059eb5a7d89b8d7174943b7a3a94e7b3aa853e3fadc6fb3bb"},"schema_version":"1.0"},"canonical_sha256":"cb7af712f8b84365d82d32cb2fb423e02cdbbf4b4b09ea61234537726fed45a0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:45.728385Z","signature_b64":"h1OOAk97nXtWm2PfutppPdbGE8I3wSgyBrqq6WYyOcuh2WlkVVPq/iA8e1KjEOXs/SGuQsZ+VfVRw09qYCuVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb7af712f8b84365d82d32cb2fb423e02cdbbf4b4b09ea61234537726fed45a0","last_reissued_at":"2026-05-18T00:42:45.727637Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:45.727637Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.02335","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":"2026-05-18T00:42:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6CMu97QprTZQwOuabY6inxvj2e+tSXjNkM4wxESXSKAaSwhX2KpFxP7iu6S+wF2cGjOVyTELDpq2HWSqEIHlCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:59:41.839860Z"},"content_sha256":"4963686660ea616a6316bb8c582f8093ca091d642a6baf7f1c5af8fb8e9df408","schema_version":"1.0","event_id":"sha256:4963686660ea616a6316bb8c582f8093ca091d642a6baf7f1c5af8fb8e9df408"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:ZN5POEXYXBBWLWBNGLFS7NBD4A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Injective chromatic number of outerplanar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Behnaz Omoomi, Mahsa Mozafari-Nia","submitted_at":"2017-06-07T18:50:14Z","abstract_excerpt":"An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ that $G$ has a $k-$injective coloring is called injective chromatic number of $G$ and denoted by $\\chi_i(G)$. In this paper, the injective chromatic number of outerplanar graphs with maximum degree $\\Delta$ and girth $g$ is studied. It is shown that for every outerplanar graph, $\\chi_i(G)\\leq \\Delta+2$, and this bound is tight. Then, it is proved that for outerplanar graphs with $\\Delta=3$, $\\chi_i(G)\\leq \\Delta+1$ and the bound is tight for outerplanar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.02335","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":"2026-05-18T00:42:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"av3vbyvDZ32W0sJabecp0FDzZ89UOeRDApQnuSJAh0MPGE4H65sNcsAQMpczO/HIe8ZNVdk+KviVM3D6b2XtCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:59:41.840224Z"},"content_sha256":"db39cad253da75c2400da54cc86201ed4b324a12f38e9e92a71e612b8434fb43","schema_version":"1.0","event_id":"sha256:db39cad253da75c2400da54cc86201ed4b324a12f38e9e92a71e612b8434fb43"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZN5POEXYXBBWLWBNGLFS7NBD4A/bundle.json","state_url":"https://pith.science/pith/ZN5POEXYXBBWLWBNGLFS7NBD4A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZN5POEXYXBBWLWBNGLFS7NBD4A/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":"2026-06-26T03:59:41Z","links":{"resolver":"https://pith.science/pith/ZN5POEXYXBBWLWBNGLFS7NBD4A","bundle":"https://pith.science/pith/ZN5POEXYXBBWLWBNGLFS7NBD4A/bundle.json","state":"https://pith.science/pith/ZN5POEXYXBBWLWBNGLFS7NBD4A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZN5POEXYXBBWLWBNGLFS7NBD4A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:ZN5POEXYXBBWLWBNGLFS7NBD4A","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":"b6fac02ec7458bd059eb5a7d89b8d7174943b7a3a94e7b3aa853e3fadc6fb3bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-07T18:50:14Z","title_canon_sha256":"de398c818f43ead7f08e4577da99a3e7ea65923e6d3a80600156f5da68e0079b"},"schema_version":"1.0","source":{"id":"1706.02335","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.02335","created_at":"2026-05-18T00:42:45Z"},{"alias_kind":"arxiv_version","alias_value":"1706.02335v1","created_at":"2026-05-18T00:42:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.02335","created_at":"2026-05-18T00:42:45Z"},{"alias_kind":"pith_short_12","alias_value":"ZN5POEXYXBBW","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZN5POEXYXBBWLWBN","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZN5POEXY","created_at":"2026-05-18T12:31:59Z"}],"graph_snapshots":[{"event_id":"sha256:db39cad253da75c2400da54cc86201ed4b324a12f38e9e92a71e612b8434fb43","target":"graph","created_at":"2026-05-18T00:42:45Z","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":"An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ that $G$ has a $k-$injective coloring is called injective chromatic number of $G$ and denoted by $\\chi_i(G)$. In this paper, the injective chromatic number of outerplanar graphs with maximum degree $\\Delta$ and girth $g$ is studied. It is shown that for every outerplanar graph, $\\chi_i(G)\\leq \\Delta+2$, and this bound is tight. Then, it is proved that for outerplanar graphs with $\\Delta=3$, $\\chi_i(G)\\leq \\Delta+1$ and the bound is tight for outerplanar","authors_text":"Behnaz Omoomi, Mahsa Mozafari-Nia","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-07T18:50:14Z","title":"Injective chromatic number of outerplanar graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.02335","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":"sha256:4963686660ea616a6316bb8c582f8093ca091d642a6baf7f1c5af8fb8e9df408","target":"record","created_at":"2026-05-18T00:42:45Z","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":"b6fac02ec7458bd059eb5a7d89b8d7174943b7a3a94e7b3aa853e3fadc6fb3bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-07T18:50:14Z","title_canon_sha256":"de398c818f43ead7f08e4577da99a3e7ea65923e6d3a80600156f5da68e0079b"},"schema_version":"1.0","source":{"id":"1706.02335","kind":"arxiv","version":1}},"canonical_sha256":"cb7af712f8b84365d82d32cb2fb423e02cdbbf4b4b09ea61234537726fed45a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb7af712f8b84365d82d32cb2fb423e02cdbbf4b4b09ea61234537726fed45a0","first_computed_at":"2026-05-18T00:42:45.727637Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:45.727637Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h1OOAk97nXtWm2PfutppPdbGE8I3wSgyBrqq6WYyOcuh2WlkVVPq/iA8e1KjEOXs/SGuQsZ+VfVRw09qYCuVCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:45.728385Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.02335","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4963686660ea616a6316bb8c582f8093ca091d642a6baf7f1c5af8fb8e9df408","sha256:db39cad253da75c2400da54cc86201ed4b324a12f38e9e92a71e612b8434fb43"],"state_sha256":"87860e01bcb4d0786f1e64eb70b0a6d85a99a62ff2d0e96825bf0b4da87bc1c6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qhKjmlTRcy1FEA7c/mCsi1P6XpzNamOKHZ2aW4KNcAwvk5zcG+dVeAiZhv4T3luhZ52I7Wp6/rf3uHagR9NSDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T03:59:41.842145Z","bundle_sha256":"97179d48d6eb9c961858dc67fd8f8cad05b157c659b9be145a5804ea182b38db"}}