{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:MTSVA66RXEATAMIZJSP6SF3UDP","short_pith_number":"pith:MTSVA66R","canonical_record":{"source":{"id":"1810.03932","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-09T12:15:12Z","cross_cats_sorted":[],"title_canon_sha256":"34c0816d5a2c20b98da75bbdac3725839ce3dd7dd7245d390a54573efca0cf1e","abstract_canon_sha256":"409fcd37c17d3e779b25b742e52529152ba781917e356b635ff7ef7e70f6a564"},"schema_version":"1.0"},"canonical_sha256":"64e5507bd1b9013031194c9fe917741bfc6e0cc8126d517b9380da716b787640","source":{"kind":"arxiv","id":"1810.03932","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.03932","created_at":"2026-05-18T00:03:44Z"},{"alias_kind":"arxiv_version","alias_value":"1810.03932v1","created_at":"2026-05-18T00:03:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.03932","created_at":"2026-05-18T00:03:44Z"},{"alias_kind":"pith_short_12","alias_value":"MTSVA66RXEAT","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"MTSVA66RXEATAMIZ","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"MTSVA66R","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:MTSVA66RXEATAMIZJSP6SF3UDP","target":"record","payload":{"canonical_record":{"source":{"id":"1810.03932","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-09T12:15:12Z","cross_cats_sorted":[],"title_canon_sha256":"34c0816d5a2c20b98da75bbdac3725839ce3dd7dd7245d390a54573efca0cf1e","abstract_canon_sha256":"409fcd37c17d3e779b25b742e52529152ba781917e356b635ff7ef7e70f6a564"},"schema_version":"1.0"},"canonical_sha256":"64e5507bd1b9013031194c9fe917741bfc6e0cc8126d517b9380da716b787640","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:44.249538Z","signature_b64":"YO3nyjLYa/GH7qknNtZodz9fvGPENNlg2VZ37PcvO0nd9HeFUXCEzER+NEiMH5ZPToQXkZH28QPxa9Y2ZAgXAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64e5507bd1b9013031194c9fe917741bfc6e0cc8126d517b9380da716b787640","last_reissued_at":"2026-05-18T00:03:44.249105Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:44.249105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.03932","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":"2026-05-18T00:03:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cBJkSueUHxXTHH2cjPT41BIg7VnRf7O7r9aocJKWnqzt3i/Go3o5Wji6mgvACK06hg9g/MnUlg5joSFQNzp/Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T17:38:51.819985Z"},"content_sha256":"dd86d6db031f3020e42f9f126502961e5cb65f21846ae2cdf7e371e4d8b01527","schema_version":"1.0","event_id":"sha256:dd86d6db031f3020e42f9f126502961e5cb65f21846ae2cdf7e371e4d8b01527"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:MTSVA66RXEATAMIZJSP6SF3UDP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Distinguishing infinite graphs with bounded degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Florian Lehner, Marcin Stawiski, Monika Pil\\'sniak","submitted_at":"2018-10-09T12:15:12Z","abstract_excerpt":"Call a colouring of a graph distinguishing, if the only colour preserving automorphism is the identity. A conjecture of Tucker states that if every automorphism of a graph $G$ moves infinitely many vertices, then there is a distinguishing $2$-colouring. We confirm this conjecture for graphs with maximum degree $\\Delta \\leq 5$. Furthermore, using similar techniques we show that if an infinite graph has maximum degree $\\Delta \\geq 3$, then it admits a distinguishing colouring with $\\Delta - 1$ colours. This bound is sharp."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.03932","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":"2026-05-18T00:03:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U7/NCd0ACpKBYC/rR+mM1afp3ofiev6K+g76CcbXp2iOQmcR7Mfaygf08XQKyokSELswfvbaK/w3VR+zVbyADA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T17:38:51.820443Z"},"content_sha256":"4455e4d8879160ae1a8bb3dad86d09aa346a0cac21b910c059eb4a81328d24ae","schema_version":"1.0","event_id":"sha256:4455e4d8879160ae1a8bb3dad86d09aa346a0cac21b910c059eb4a81328d24ae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MTSVA66RXEATAMIZJSP6SF3UDP/bundle.json","state_url":"https://pith.science/pith/MTSVA66RXEATAMIZJSP6SF3UDP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MTSVA66RXEATAMIZJSP6SF3UDP/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":"2026-06-23T17:38:51Z","links":{"resolver":"https://pith.science/pith/MTSVA66RXEATAMIZJSP6SF3UDP","bundle":"https://pith.science/pith/MTSVA66RXEATAMIZJSP6SF3UDP/bundle.json","state":"https://pith.science/pith/MTSVA66RXEATAMIZJSP6SF3UDP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MTSVA66RXEATAMIZJSP6SF3UDP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:MTSVA66RXEATAMIZJSP6SF3UDP","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":"409fcd37c17d3e779b25b742e52529152ba781917e356b635ff7ef7e70f6a564","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-09T12:15:12Z","title_canon_sha256":"34c0816d5a2c20b98da75bbdac3725839ce3dd7dd7245d390a54573efca0cf1e"},"schema_version":"1.0","source":{"id":"1810.03932","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.03932","created_at":"2026-05-18T00:03:44Z"},{"alias_kind":"arxiv_version","alias_value":"1810.03932v1","created_at":"2026-05-18T00:03:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.03932","created_at":"2026-05-18T00:03:44Z"},{"alias_kind":"pith_short_12","alias_value":"MTSVA66RXEAT","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"MTSVA66RXEATAMIZ","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"MTSVA66R","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:4455e4d8879160ae1a8bb3dad86d09aa346a0cac21b910c059eb4a81328d24ae","target":"graph","created_at":"2026-05-18T00:03:44Z","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":"Call a colouring of a graph distinguishing, if the only colour preserving automorphism is the identity. A conjecture of Tucker states that if every automorphism of a graph $G$ moves infinitely many vertices, then there is a distinguishing $2$-colouring. We confirm this conjecture for graphs with maximum degree $\\Delta \\leq 5$. Furthermore, using similar techniques we show that if an infinite graph has maximum degree $\\Delta \\geq 3$, then it admits a distinguishing colouring with $\\Delta - 1$ colours. This bound is sharp.","authors_text":"Florian Lehner, Marcin Stawiski, Monika Pil\\'sniak","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-09T12:15:12Z","title":"Distinguishing infinite graphs with bounded degrees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.03932","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:dd86d6db031f3020e42f9f126502961e5cb65f21846ae2cdf7e371e4d8b01527","target":"record","created_at":"2026-05-18T00:03:44Z","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":"409fcd37c17d3e779b25b742e52529152ba781917e356b635ff7ef7e70f6a564","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-09T12:15:12Z","title_canon_sha256":"34c0816d5a2c20b98da75bbdac3725839ce3dd7dd7245d390a54573efca0cf1e"},"schema_version":"1.0","source":{"id":"1810.03932","kind":"arxiv","version":1}},"canonical_sha256":"64e5507bd1b9013031194c9fe917741bfc6e0cc8126d517b9380da716b787640","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64e5507bd1b9013031194c9fe917741bfc6e0cc8126d517b9380da716b787640","first_computed_at":"2026-05-18T00:03:44.249105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:03:44.249105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YO3nyjLYa/GH7qknNtZodz9fvGPENNlg2VZ37PcvO0nd9HeFUXCEzER+NEiMH5ZPToQXkZH28QPxa9Y2ZAgXAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:03:44.249538Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.03932","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd86d6db031f3020e42f9f126502961e5cb65f21846ae2cdf7e371e4d8b01527","sha256:4455e4d8879160ae1a8bb3dad86d09aa346a0cac21b910c059eb4a81328d24ae"],"state_sha256":"1941a3312e1d69cd258a8b2c15d52441b56241387e1aa598d9d4f199d3abc886"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SQhdUizgbstwcfXJHq4TfMkaUu5m6PmMCBSzdGhe5hp3doixir3DElZoTXo7KZNSbkaGmWmzT9m0GXuahJbLAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T17:38:51.823154Z","bundle_sha256":"bb89e5117bd6e1cd70743ad531d729902ee4f248f62650cd19b23bcd5dfb95f5"}}