{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:4OAJ23GJ4VFLMRWDXJNUNWMSFZ","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":"796e621c88c7e2f554e67f694cd2b2b39e43ba4a5ce4f809afae97dd04380db5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-23T08:41:25Z","title_canon_sha256":"daa6f6335d6656666c2ec064cad5a6757b4635f4cc1a0d5d0601a50f3a63cde4"},"schema_version":"1.0","source":{"id":"1509.06884","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.06884","created_at":"2026-05-18T01:32:17Z"},{"alias_kind":"arxiv_version","alias_value":"1509.06884v1","created_at":"2026-05-18T01:32:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06884","created_at":"2026-05-18T01:32:17Z"},{"alias_kind":"pith_short_12","alias_value":"4OAJ23GJ4VFL","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4OAJ23GJ4VFLMRWD","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4OAJ23GJ","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:09ff76ecc1751fb68668bfa117d8efba92c77e107390d7b64a6a72295fd79ea4","target":"graph","created_at":"2026-05-18T01:32:17Z","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":"This paper introduces a new variant of hypercubes, which we call Z-cubes. The n-dimensional Z-cube $H_n$ is obtained from two copies of the (n-1)-dimensional Z-cube $H_{n-1}$ by adding a special perfect matching between the vertices of these two copies of $H_{n-1}$. We prove that the n-dimensional Z-cubes $H_n$ has diameter $(1+o(1))n/\\log_2 n$. This greatly improves on the previous known variants of hypercube of dimension n, whose diameters are all larger than n/3. Moreover, any hypercube variant of dimension $n$ is an n-regular graph on $2^n$ vertices, and hence has diameter greater than $n/","authors_text":"Xuding Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-23T08:41:25Z","title":"The Z-cubes: a hypercube variant with small diameter"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06884","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":"sha256:728d37a0271792d2336f1046d90b64b4ff6ee1a7264ca61684f923e02fab8253","target":"record","created_at":"2026-05-18T01:32:17Z","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":"796e621c88c7e2f554e67f694cd2b2b39e43ba4a5ce4f809afae97dd04380db5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-23T08:41:25Z","title_canon_sha256":"daa6f6335d6656666c2ec064cad5a6757b4635f4cc1a0d5d0601a50f3a63cde4"},"schema_version":"1.0","source":{"id":"1509.06884","kind":"arxiv","version":1}},"canonical_sha256":"e3809d6cc9e54ab646c3ba5b46d9922e699b8107cbdd5fcca4e4481127ca7f33","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3809d6cc9e54ab646c3ba5b46d9922e699b8107cbdd5fcca4e4481127ca7f33","first_computed_at":"2026-05-18T01:32:17.647492Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:17.647492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yXmWTnwu2sOKUzg0tCLpehsVZZMV4cbyxdB/mf5YspKePzdkDWmwjadWGMsumKUkqjBO9aYZI+bjFglraaWEAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:17.648247Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.06884","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:728d37a0271792d2336f1046d90b64b4ff6ee1a7264ca61684f923e02fab8253","sha256:09ff76ecc1751fb68668bfa117d8efba92c77e107390d7b64a6a72295fd79ea4"],"state_sha256":"2caeed1f4b193e9a4060a70f4937509c1da3b5b7c978595f942ed4a1b8e85082"}