{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:MGMNNPI3I5WFBSCJGGOTQXUXK5","short_pith_number":"pith:MGMNNPI3","canonical_record":{"source":{"id":"1707.06763","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-07-21T05:52:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4d96dc47fcc1c25fbff460740731ef8c29480a84dcf7b799e284ebac64dc462f","abstract_canon_sha256":"a39f31a2b57995e612c615b1f07e58fb39438d52000cc42247dabf911e86f102"},"schema_version":"1.0"},"canonical_sha256":"6198d6bd1b476c50c849319d385e975761fd0931cf34a5437670dbfb65e4c921","source":{"kind":"arxiv","id":"1707.06763","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.06763","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"arxiv_version","alias_value":"1707.06763v2","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.06763","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"pith_short_12","alias_value":"MGMNNPI3I5WF","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"MGMNNPI3I5WFBSCJ","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"MGMNNPI3","created_at":"2026-05-18T12:31:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:MGMNNPI3I5WFBSCJGGOTQXUXK5","target":"record","payload":{"canonical_record":{"source":{"id":"1707.06763","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-07-21T05:52:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4d96dc47fcc1c25fbff460740731ef8c29480a84dcf7b799e284ebac64dc462f","abstract_canon_sha256":"a39f31a2b57995e612c615b1f07e58fb39438d52000cc42247dabf911e86f102"},"schema_version":"1.0"},"canonical_sha256":"6198d6bd1b476c50c849319d385e975761fd0931cf34a5437670dbfb65e4c921","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:11.787769Z","signature_b64":"JvEhQJjnwJ10x64JA3Hx3p/jKPiDlBgNREmaUczpfO7Q8qnuiUk8OHDDSbiQkGSOZ6DhbKEe797tMZ/mZ4arDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6198d6bd1b476c50c849319d385e975761fd0931cf34a5437670dbfb65e4c921","last_reissued_at":"2026-05-18T00:39:11.787068Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:11.787068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.06763","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":"2026-05-18T00:39:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6mXoYMSfQ/7KeFrdSL/HqBm7W6da6sm70/UsYPl8E3Qk884ci/FBm9OgLyCdVdXof9rqtVfEKiK4kJAumXM8Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T20:57:57.155806Z"},"content_sha256":"497704dd7a8f5f7eb207b31d364fa2a6c5f575bcb686144b248719c2f74d858e","schema_version":"1.0","event_id":"sha256:497704dd7a8f5f7eb207b31d364fa2a6c5f575bcb686144b248719c2f74d858e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:MGMNNPI3I5WFBSCJGGOTQXUXK5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Orbits of Crossed Cubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Kung-Jui Pai, Tzong-Huei Shiau, Yue-Li Wang","submitted_at":"2017-07-21T05:52:17Z","abstract_excerpt":"An orbit of $G$ is a subset $S$ of $V(G)$ such that $\\phi(u)=v$ for any two vertices $u,v\\in S$, where $\\phi$ is an isomorphism of $G$. The orbit number of a graph $G$, denoted by $\\text{Orb}(G)$, is the number of orbits of $G$. In [A Note on Path Embedding in Crossed Cubes with Faulty Vertices, Information Processing Letters 121 (2017) pp. 34--38], Chen et al. conjectured that $\\text{Orb}(\\text{CQ}_n)=2^{\\lceil\\frac{n}{2}\\rceil-2}$ for $n\\geqslant 3$, where $\\text{CQ}_n$ denotes an $n$-dimensional crossed cube. In this paper, we settle the conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06763","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":"2026-05-18T00:39:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RUHI4XNY2hkOL8jJFQ6LBPba15LsT55QfOnAeJtxC9146JvHA/O+ht2VUwuPzpU7yuQOgovsxBnjSin6zv7qAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T20:57:57.156154Z"},"content_sha256":"a870549edf720c74dff054802f6ad08239bfc129f9c86364058a432912e12d41","schema_version":"1.0","event_id":"sha256:a870549edf720c74dff054802f6ad08239bfc129f9c86364058a432912e12d41"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MGMNNPI3I5WFBSCJGGOTQXUXK5/bundle.json","state_url":"https://pith.science/pith/MGMNNPI3I5WFBSCJGGOTQXUXK5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MGMNNPI3I5WFBSCJGGOTQXUXK5/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":"2026-07-04T20:57:57Z","links":{"resolver":"https://pith.science/pith/MGMNNPI3I5WFBSCJGGOTQXUXK5","bundle":"https://pith.science/pith/MGMNNPI3I5WFBSCJGGOTQXUXK5/bundle.json","state":"https://pith.science/pith/MGMNNPI3I5WFBSCJGGOTQXUXK5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MGMNNPI3I5WFBSCJGGOTQXUXK5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:MGMNNPI3I5WFBSCJGGOTQXUXK5","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":"a39f31a2b57995e612c615b1f07e58fb39438d52000cc42247dabf911e86f102","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-07-21T05:52:17Z","title_canon_sha256":"4d96dc47fcc1c25fbff460740731ef8c29480a84dcf7b799e284ebac64dc462f"},"schema_version":"1.0","source":{"id":"1707.06763","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.06763","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"arxiv_version","alias_value":"1707.06763v2","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.06763","created_at":"2026-05-18T00:39:11Z"},{"alias_kind":"pith_short_12","alias_value":"MGMNNPI3I5WF","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"MGMNNPI3I5WFBSCJ","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"MGMNNPI3","created_at":"2026-05-18T12:31:31Z"}],"graph_snapshots":[{"event_id":"sha256:a870549edf720c74dff054802f6ad08239bfc129f9c86364058a432912e12d41","target":"graph","created_at":"2026-05-18T00:39:11Z","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 orbit of $G$ is a subset $S$ of $V(G)$ such that $\\phi(u)=v$ for any two vertices $u,v\\in S$, where $\\phi$ is an isomorphism of $G$. The orbit number of a graph $G$, denoted by $\\text{Orb}(G)$, is the number of orbits of $G$. In [A Note on Path Embedding in Crossed Cubes with Faulty Vertices, Information Processing Letters 121 (2017) pp. 34--38], Chen et al. conjectured that $\\text{Orb}(\\text{CQ}_n)=2^{\\lceil\\frac{n}{2}\\rceil-2}$ for $n\\geqslant 3$, where $\\text{CQ}_n$ denotes an $n$-dimensional crossed cube. In this paper, we settle the conjecture.","authors_text":"Kung-Jui Pai, Tzong-Huei Shiau, Yue-Li Wang","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-07-21T05:52:17Z","title":"On the Orbits of Crossed Cubes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06763","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":"sha256:497704dd7a8f5f7eb207b31d364fa2a6c5f575bcb686144b248719c2f74d858e","target":"record","created_at":"2026-05-18T00:39:11Z","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":"a39f31a2b57995e612c615b1f07e58fb39438d52000cc42247dabf911e86f102","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-07-21T05:52:17Z","title_canon_sha256":"4d96dc47fcc1c25fbff460740731ef8c29480a84dcf7b799e284ebac64dc462f"},"schema_version":"1.0","source":{"id":"1707.06763","kind":"arxiv","version":2}},"canonical_sha256":"6198d6bd1b476c50c849319d385e975761fd0931cf34a5437670dbfb65e4c921","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6198d6bd1b476c50c849319d385e975761fd0931cf34a5437670dbfb65e4c921","first_computed_at":"2026-05-18T00:39:11.787068Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:11.787068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JvEhQJjnwJ10x64JA3Hx3p/jKPiDlBgNREmaUczpfO7Q8qnuiUk8OHDDSbiQkGSOZ6DhbKEe797tMZ/mZ4arDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:11.787769Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.06763","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:497704dd7a8f5f7eb207b31d364fa2a6c5f575bcb686144b248719c2f74d858e","sha256:a870549edf720c74dff054802f6ad08239bfc129f9c86364058a432912e12d41"],"state_sha256":"8f0a0305aca008aa91081eb4f6036806c09e6154a1b2620b8a2049f1f77ea8d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hxUPo1WNWvivJ1Nvb+khok09dLpdF3Yh+yywFJKoaUtAU2QE4PqTiaJZsl/Q/3L+m75ndrM+/KWg9bNdHeoEBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T20:57:57.158093Z","bundle_sha256":"2d1b5526452d4d18c5347ec044c78c683e7cce5dacf6f9c3cfb69ba1dabf722b"}}