{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:JFN3NMPOA4IIC4DRMFROENRJXV","short_pith_number":"pith:JFN3NMPO","canonical_record":{"source":{"id":"1410.8328","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2014-10-30T11:14:30Z","cross_cats_sorted":[],"title_canon_sha256":"ffbc931740ab19b5c3293a18037bd4fd8636a0666b4dce503d101f15697e4a49","abstract_canon_sha256":"e51be8dcebdcc32d115252646ecb29395b7656407a3aac6096e709b588b79179"},"schema_version":"1.0"},"canonical_sha256":"495bb6b1ee07108170716162e23629bd7c93ac8684523411347d915c9a261f56","source":{"kind":"arxiv","id":"1410.8328","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.8328","created_at":"2026-05-18T02:39:00Z"},{"alias_kind":"arxiv_version","alias_value":"1410.8328v1","created_at":"2026-05-18T02:39:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.8328","created_at":"2026-05-18T02:39:00Z"},{"alias_kind":"pith_short_12","alias_value":"JFN3NMPOA4II","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"JFN3NMPOA4IIC4DR","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"JFN3NMPO","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:JFN3NMPOA4IIC4DRMFROENRJXV","target":"record","payload":{"canonical_record":{"source":{"id":"1410.8328","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2014-10-30T11:14:30Z","cross_cats_sorted":[],"title_canon_sha256":"ffbc931740ab19b5c3293a18037bd4fd8636a0666b4dce503d101f15697e4a49","abstract_canon_sha256":"e51be8dcebdcc32d115252646ecb29395b7656407a3aac6096e709b588b79179"},"schema_version":"1.0"},"canonical_sha256":"495bb6b1ee07108170716162e23629bd7c93ac8684523411347d915c9a261f56","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:00.134601Z","signature_b64":"M6rKsCLoCC54Zj1EkHqGhckAyWJhcSK5AEoEcoCpAqf74WWZmCYg1hGaOrwX7iK9yzSN1F8zGEeK4WKSm0GyDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"495bb6b1ee07108170716162e23629bd7c93ac8684523411347d915c9a261f56","last_reissued_at":"2026-05-18T02:39:00.134097Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:00.134097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.8328","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":"2026-05-18T02:39:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gAYRa0fE5muuCRHzVgmlbZJXLz7Ocrg6ryA6xokQd/W8kiG9y8v3RkwE3kwWRyR8hBDYGafOOdfCn84RdAcYCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T22:06:41.149172Z"},"content_sha256":"8333e74af05d9c21f35974e423cea721aa1c1eb3969ab26d31fbb1c28ba295d0","schema_version":"1.0","event_id":"sha256:8333e74af05d9c21f35974e423cea721aa1c1eb3969ab26d31fbb1c28ba295d0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:JFN3NMPOA4IIC4DRMFROENRJXV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Contemplating some invariants of the Jaco Graph, $J_n(1), n \\in \\Bbb N$","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Johan Kok, Susanth C","submitted_at":"2014-10-30T11:14:30Z","abstract_excerpt":"Kok et.al. [7] introduced Jaco Graphs (\\emph{order 1}). In this essay we present a recursive formula to determine the \\emph{independence number} $\\alpha(J_n(1)) = |\\Bbb I|$ with, $\\Bbb I = \\{v_{i,j}| v_1 = v_{1,1} \\in \\Bbb I$ and $v_i = v_{i,j} =v_{(d^+(v_{m, (j-1)}) + m +1)}\\}.$ We also prove that for the Jaco Graph, $J_n(1), n \\in \\Bbb N$ with the prime Jaconian vertex $v_i$ the chromatic number, $\\chi(J_n(1))$ is given by: \\begin{equation*} \\chi(J_n(1)) \\begin{cases} = (n-i) + 1, &\\text{if and only if the edge $v_iv_n$ exists,}\\\\ \\\\ = n-i &\\text{otherwise.} \\end{cases} \\end{equation*} We fu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8328","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":"2026-05-18T02:39:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IBtru6Hl4JE6C07h2P5FkTUsnc6jFtfGX+LraPl9hNbDdC5HYLUr9moZV96/Id9LbRvWi7zr3UERXMgQ12KoCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T22:06:41.149531Z"},"content_sha256":"219c194541dbe789e73f2951e306ce353f3b866557db06b481639343a5dc5d80","schema_version":"1.0","event_id":"sha256:219c194541dbe789e73f2951e306ce353f3b866557db06b481639343a5dc5d80"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV/bundle.json","state_url":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JFN3NMPOA4IIC4DRMFROENRJXV/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":"2026-06-03T22:06:41Z","links":{"resolver":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV","bundle":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV/bundle.json","state":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JFN3NMPOA4IIC4DRMFROENRJXV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:JFN3NMPOA4IIC4DRMFROENRJXV","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":"e51be8dcebdcc32d115252646ecb29395b7656407a3aac6096e709b588b79179","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2014-10-30T11:14:30Z","title_canon_sha256":"ffbc931740ab19b5c3293a18037bd4fd8636a0666b4dce503d101f15697e4a49"},"schema_version":"1.0","source":{"id":"1410.8328","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.8328","created_at":"2026-05-18T02:39:00Z"},{"alias_kind":"arxiv_version","alias_value":"1410.8328v1","created_at":"2026-05-18T02:39:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.8328","created_at":"2026-05-18T02:39:00Z"},{"alias_kind":"pith_short_12","alias_value":"JFN3NMPOA4II","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"JFN3NMPOA4IIC4DR","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"JFN3NMPO","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:219c194541dbe789e73f2951e306ce353f3b866557db06b481639343a5dc5d80","target":"graph","created_at":"2026-05-18T02:39:00Z","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":"Kok et.al. [7] introduced Jaco Graphs (\\emph{order 1}). In this essay we present a recursive formula to determine the \\emph{independence number} $\\alpha(J_n(1)) = |\\Bbb I|$ with, $\\Bbb I = \\{v_{i,j}| v_1 = v_{1,1} \\in \\Bbb I$ and $v_i = v_{i,j} =v_{(d^+(v_{m, (j-1)}) + m +1)}\\}.$ We also prove that for the Jaco Graph, $J_n(1), n \\in \\Bbb N$ with the prime Jaconian vertex $v_i$ the chromatic number, $\\chi(J_n(1))$ is given by: \\begin{equation*} \\chi(J_n(1)) \\begin{cases} = (n-i) + 1, &\\text{if and only if the edge $v_iv_n$ exists,}\\\\ \\\\ = n-i &\\text{otherwise.} \\end{cases} \\end{equation*} We fu","authors_text":"Johan Kok, Susanth C","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2014-10-30T11:14:30Z","title":"Contemplating some invariants of the Jaco Graph, $J_n(1), n \\in \\Bbb N$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8328","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":"sha256:8333e74af05d9c21f35974e423cea721aa1c1eb3969ab26d31fbb1c28ba295d0","target":"record","created_at":"2026-05-18T02:39:00Z","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":"e51be8dcebdcc32d115252646ecb29395b7656407a3aac6096e709b588b79179","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2014-10-30T11:14:30Z","title_canon_sha256":"ffbc931740ab19b5c3293a18037bd4fd8636a0666b4dce503d101f15697e4a49"},"schema_version":"1.0","source":{"id":"1410.8328","kind":"arxiv","version":1}},"canonical_sha256":"495bb6b1ee07108170716162e23629bd7c93ac8684523411347d915c9a261f56","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"495bb6b1ee07108170716162e23629bd7c93ac8684523411347d915c9a261f56","first_computed_at":"2026-05-18T02:39:00.134097Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:39:00.134097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M6rKsCLoCC54Zj1EkHqGhckAyWJhcSK5AEoEcoCpAqf74WWZmCYg1hGaOrwX7iK9yzSN1F8zGEeK4WKSm0GyDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:39:00.134601Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.8328","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8333e74af05d9c21f35974e423cea721aa1c1eb3969ab26d31fbb1c28ba295d0","sha256:219c194541dbe789e73f2951e306ce353f3b866557db06b481639343a5dc5d80"],"state_sha256":"35d05b4fb182c7f26da587cf3bf22d3e89402c9cc625f79e7d211f87983a8adc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1xYGjW/sy1fWuhTbkqB9UFtPMHki7Th5JK19ygg8PY5zwJAT95zDkZDErrl0IZbLFQjdKycj7XPOmS5Ihj6PBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T22:06:41.151532Z","bundle_sha256":"45e4309f80e0a372d11e9beb418d2fbfb8ce926a96d4fa8b99b2327218f52eb3"}}