{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:3RF7RRE7PWSPS57GDZW4B6UGWB","short_pith_number":"pith:3RF7RRE7","canonical_record":{"source":{"id":"1702.03527","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-12T13:36:10Z","cross_cats_sorted":[],"title_canon_sha256":"634184068640963cfa1073256696384ca075b780d5e6aa03d39cc707ffd3fecf","abstract_canon_sha256":"d634cb7df838fb8d98ef50811175bec25b3e8279d95b319cbdb846a4947af2eb"},"schema_version":"1.0"},"canonical_sha256":"dc4bf8c49f7da4f977e61e6dc0fa86b0542f26291c260ea1a0f455c88d27f5fe","source":{"kind":"arxiv","id":"1702.03527","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.03527","created_at":"2026-05-18T00:50:53Z"},{"alias_kind":"arxiv_version","alias_value":"1702.03527v1","created_at":"2026-05-18T00:50:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.03527","created_at":"2026-05-18T00:50:53Z"},{"alias_kind":"pith_short_12","alias_value":"3RF7RRE7PWSP","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3RF7RRE7PWSPS57G","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3RF7RRE7","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:3RF7RRE7PWSPS57GDZW4B6UGWB","target":"record","payload":{"canonical_record":{"source":{"id":"1702.03527","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-12T13:36:10Z","cross_cats_sorted":[],"title_canon_sha256":"634184068640963cfa1073256696384ca075b780d5e6aa03d39cc707ffd3fecf","abstract_canon_sha256":"d634cb7df838fb8d98ef50811175bec25b3e8279d95b319cbdb846a4947af2eb"},"schema_version":"1.0"},"canonical_sha256":"dc4bf8c49f7da4f977e61e6dc0fa86b0542f26291c260ea1a0f455c88d27f5fe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:53.573372Z","signature_b64":"kRXjPvmIFVEL8YXxj6/c0xfNAd9X++wAqEEFe2guJYkb+jDH+VK1np3r9ATSmIsts9DlP4RtRP1Wb878IHLoBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc4bf8c49f7da4f977e61e6dc0fa86b0542f26291c260ea1a0f455c88d27f5fe","last_reissued_at":"2026-05-18T00:50:53.572812Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:53.572812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1702.03527","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":"2026-05-18T00:50:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qy/n9OAzS8MFXc+jXuDQY1KddY+plTsIfXibu3w9Ku6eMve29dvC01GCSOglpvUa1FIUPlx74aMGxb84lhsIDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T02:33:27.020022Z"},"content_sha256":"04a27d0cf4948466e5fa994d4e510c49a794310d264fdec8854561ea9ded592c","schema_version":"1.0","event_id":"sha256:04a27d0cf4948466e5fa994d4e510c49a794310d264fdec8854561ea9ded592c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:3RF7RRE7PWSPS57GDZW4B6UGWB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Connectedness of Certain Graph Coloring Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nandini Nilakantan, Samir Shukla","submitted_at":"2017-02-12T13:36:10Z","abstract_excerpt":"In this article, we consider the bipartite graphs $K_2 \\times K_n$. We prove that the connectedness of the complex $\\displaystyle \\text{Hom}(K_2\\times K_{n}, K_m) $ is $m-n-1$ if $m \\geq n$ and $m-3$ in the other cases. Therefore, we show that for this class of graphs, $\\text{Hom} (G, K_m)$ is exactly $m-d-2$ connected, $m \\geq n$, where $d$ is the maximal degree of the graph $G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.03527","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":"2026-05-18T00:50:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nldz0EqnGWgl6VOiBv9DV4clwegG6Qra2J4hfBOw1QJ2ITLF3VBwyTOtE6LZuhYI0BmX5Zt7lKPeV1ni0m3IBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T02:33:27.020439Z"},"content_sha256":"86954933aae046e8d081f48204d74f22c97e5da28daa7b4a21f61f20d9feba41","schema_version":"1.0","event_id":"sha256:86954933aae046e8d081f48204d74f22c97e5da28daa7b4a21f61f20d9feba41"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3RF7RRE7PWSPS57GDZW4B6UGWB/bundle.json","state_url":"https://pith.science/pith/3RF7RRE7PWSPS57GDZW4B6UGWB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3RF7RRE7PWSPS57GDZW4B6UGWB/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":"2026-05-26T02:33:27Z","links":{"resolver":"https://pith.science/pith/3RF7RRE7PWSPS57GDZW4B6UGWB","bundle":"https://pith.science/pith/3RF7RRE7PWSPS57GDZW4B6UGWB/bundle.json","state":"https://pith.science/pith/3RF7RRE7PWSPS57GDZW4B6UGWB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3RF7RRE7PWSPS57GDZW4B6UGWB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:3RF7RRE7PWSPS57GDZW4B6UGWB","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":"d634cb7df838fb8d98ef50811175bec25b3e8279d95b319cbdb846a4947af2eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-12T13:36:10Z","title_canon_sha256":"634184068640963cfa1073256696384ca075b780d5e6aa03d39cc707ffd3fecf"},"schema_version":"1.0","source":{"id":"1702.03527","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.03527","created_at":"2026-05-18T00:50:53Z"},{"alias_kind":"arxiv_version","alias_value":"1702.03527v1","created_at":"2026-05-18T00:50:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.03527","created_at":"2026-05-18T00:50:53Z"},{"alias_kind":"pith_short_12","alias_value":"3RF7RRE7PWSP","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3RF7RRE7PWSPS57G","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3RF7RRE7","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:86954933aae046e8d081f48204d74f22c97e5da28daa7b4a21f61f20d9feba41","target":"graph","created_at":"2026-05-18T00:50:53Z","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":"In this article, we consider the bipartite graphs $K_2 \\times K_n$. We prove that the connectedness of the complex $\\displaystyle \\text{Hom}(K_2\\times K_{n}, K_m) $ is $m-n-1$ if $m \\geq n$ and $m-3$ in the other cases. Therefore, we show that for this class of graphs, $\\text{Hom} (G, K_m)$ is exactly $m-d-2$ connected, $m \\geq n$, where $d$ is the maximal degree of the graph $G$.","authors_text":"Nandini Nilakantan, Samir Shukla","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-12T13:36:10Z","title":"Connectedness of Certain Graph Coloring Complexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.03527","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":"sha256:04a27d0cf4948466e5fa994d4e510c49a794310d264fdec8854561ea9ded592c","target":"record","created_at":"2026-05-18T00:50:53Z","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":"d634cb7df838fb8d98ef50811175bec25b3e8279d95b319cbdb846a4947af2eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-12T13:36:10Z","title_canon_sha256":"634184068640963cfa1073256696384ca075b780d5e6aa03d39cc707ffd3fecf"},"schema_version":"1.0","source":{"id":"1702.03527","kind":"arxiv","version":1}},"canonical_sha256":"dc4bf8c49f7da4f977e61e6dc0fa86b0542f26291c260ea1a0f455c88d27f5fe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dc4bf8c49f7da4f977e61e6dc0fa86b0542f26291c260ea1a0f455c88d27f5fe","first_computed_at":"2026-05-18T00:50:53.572812Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:53.572812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kRXjPvmIFVEL8YXxj6/c0xfNAd9X++wAqEEFe2guJYkb+jDH+VK1np3r9ATSmIsts9DlP4RtRP1Wb878IHLoBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:53.573372Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.03527","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:04a27d0cf4948466e5fa994d4e510c49a794310d264fdec8854561ea9ded592c","sha256:86954933aae046e8d081f48204d74f22c97e5da28daa7b4a21f61f20d9feba41"],"state_sha256":"0d8f33714e414898966ab17c88820dab04f7f95bd2ef5d3d940b6a02f4d36009"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CPsGyzlmLPmcfTig0zUmqly4vAC/TaMgWpsRLb5CU7rXeeA7zp36U8O7fRAV/GUHCLQF0DowIVDwF10rWpBxCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T02:33:27.023495Z","bundle_sha256":"14e96d63354407615d6e02da919b2cba672bd23ea7a7158bdbb78adf2904a990"}}