{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:PCTR3OP4VIJELGJZDKDCAI3X6K","short_pith_number":"pith:PCTR3OP4","canonical_record":{"source":{"id":"1610.04874","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-16T15:48:32Z","cross_cats_sorted":[],"title_canon_sha256":"0608f8c519c09565d81f32408d9ba5d400f62af491634283107fd1c98f3b1b30","abstract_canon_sha256":"1f40be69a16a1486f179d81db1a251f68174aa155282b835e9abaf2cfb918752"},"schema_version":"1.0"},"canonical_sha256":"78a71db9fcaa124599391a86202377f297350e714320a189b502512acf61b40c","source":{"kind":"arxiv","id":"1610.04874","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04874","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04874v2","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04874","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"pith_short_12","alias_value":"PCTR3OP4VIJE","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PCTR3OP4VIJELGJZ","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PCTR3OP4","created_at":"2026-05-18T12:30:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:PCTR3OP4VIJELGJZDKDCAI3X6K","target":"record","payload":{"canonical_record":{"source":{"id":"1610.04874","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-16T15:48:32Z","cross_cats_sorted":[],"title_canon_sha256":"0608f8c519c09565d81f32408d9ba5d400f62af491634283107fd1c98f3b1b30","abstract_canon_sha256":"1f40be69a16a1486f179d81db1a251f68174aa155282b835e9abaf2cfb918752"},"schema_version":"1.0"},"canonical_sha256":"78a71db9fcaa124599391a86202377f297350e714320a189b502512acf61b40c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:58.569239Z","signature_b64":"2esRmleQcpApLhS4ob6IBQDWgnrdQ19oE10c60dS7qUjvUQ8HRAlU5VBnQKJqrii2NSaPBGeZi7wbN20PVgYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78a71db9fcaa124599391a86202377f297350e714320a189b502512acf61b40c","last_reissued_at":"2026-05-18T00:45:58.568784Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:58.568784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.04874","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":"2026-05-18T00:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"is/O3kHMJytoIDCQGFkNIeK8yLUrGWtm5ZmM8Pr/tJz2Z9Lw6XL7JADT6XRbbD5Qhcrsh3BPnP8AMOUvQIxDDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T07:35:17.414493Z"},"content_sha256":"29eecb9d11ad518fd3e4c5f3d93271cd4a9875817f2733f7c1c6d0af4f756b1c","schema_version":"1.0","event_id":"sha256:29eecb9d11ad518fd3e4c5f3d93271cd4a9875817f2733f7c1c6d0af4f756b1c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:PCTR3OP4VIJELGJZDKDCAI3X6K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Coloring Graphs to Produce Properly Colored Walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Robert Melville, Wayne Goddard","submitted_at":"2016-10-16T15:48:32Z","abstract_excerpt":"For a connected graph, we define the proper-walk connection number as the minimum number of colors needed to color the edges of a graph so that there is a walk between every pair of vertices without two consecutive edges having the same color. We show that the proper-walk connection number is at most three for all cyclic graphs, and at most two for bridgeless graphs. We also characterize the bipartite graphs that have proper-walk connection number equal to two, and show that this characterization also holds for the analogous problem where one is restricted to properly colored paths."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04874","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":"2026-05-18T00:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aMoIiB1CiTASQHumHc433eCJjqjDL4b+p8KU4SrQ5hxhOqA3jpQ9E1ns1bG/cw/OnFBbsieYHpCBILnusKDRDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T07:35:17.414955Z"},"content_sha256":"73cc2b6efb39f4b9471d86a47d16a5376e1be28f368641731958004e5aa63fed","schema_version":"1.0","event_id":"sha256:73cc2b6efb39f4b9471d86a47d16a5376e1be28f368641731958004e5aa63fed"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PCTR3OP4VIJELGJZDKDCAI3X6K/bundle.json","state_url":"https://pith.science/pith/PCTR3OP4VIJELGJZDKDCAI3X6K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PCTR3OP4VIJELGJZDKDCAI3X6K/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":"2026-06-25T07:35:17Z","links":{"resolver":"https://pith.science/pith/PCTR3OP4VIJELGJZDKDCAI3X6K","bundle":"https://pith.science/pith/PCTR3OP4VIJELGJZDKDCAI3X6K/bundle.json","state":"https://pith.science/pith/PCTR3OP4VIJELGJZDKDCAI3X6K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PCTR3OP4VIJELGJZDKDCAI3X6K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:PCTR3OP4VIJELGJZDKDCAI3X6K","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":"1f40be69a16a1486f179d81db1a251f68174aa155282b835e9abaf2cfb918752","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-16T15:48:32Z","title_canon_sha256":"0608f8c519c09565d81f32408d9ba5d400f62af491634283107fd1c98f3b1b30"},"schema_version":"1.0","source":{"id":"1610.04874","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04874","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04874v2","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04874","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"pith_short_12","alias_value":"PCTR3OP4VIJE","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PCTR3OP4VIJELGJZ","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PCTR3OP4","created_at":"2026-05-18T12:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:73cc2b6efb39f4b9471d86a47d16a5376e1be28f368641731958004e5aa63fed","target":"graph","created_at":"2026-05-18T00:45:58Z","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":"For a connected graph, we define the proper-walk connection number as the minimum number of colors needed to color the edges of a graph so that there is a walk between every pair of vertices without two consecutive edges having the same color. We show that the proper-walk connection number is at most three for all cyclic graphs, and at most two for bridgeless graphs. We also characterize the bipartite graphs that have proper-walk connection number equal to two, and show that this characterization also holds for the analogous problem where one is restricted to properly colored paths.","authors_text":"Robert Melville, Wayne Goddard","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-16T15:48:32Z","title":"Coloring Graphs to Produce Properly Colored Walks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04874","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":"sha256:29eecb9d11ad518fd3e4c5f3d93271cd4a9875817f2733f7c1c6d0af4f756b1c","target":"record","created_at":"2026-05-18T00:45:58Z","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":"1f40be69a16a1486f179d81db1a251f68174aa155282b835e9abaf2cfb918752","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-16T15:48:32Z","title_canon_sha256":"0608f8c519c09565d81f32408d9ba5d400f62af491634283107fd1c98f3b1b30"},"schema_version":"1.0","source":{"id":"1610.04874","kind":"arxiv","version":2}},"canonical_sha256":"78a71db9fcaa124599391a86202377f297350e714320a189b502512acf61b40c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"78a71db9fcaa124599391a86202377f297350e714320a189b502512acf61b40c","first_computed_at":"2026-05-18T00:45:58.568784Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:58.568784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2esRmleQcpApLhS4ob6IBQDWgnrdQ19oE10c60dS7qUjvUQ8HRAlU5VBnQKJqrii2NSaPBGeZi7wbN20PVgYDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:58.569239Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.04874","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29eecb9d11ad518fd3e4c5f3d93271cd4a9875817f2733f7c1c6d0af4f756b1c","sha256:73cc2b6efb39f4b9471d86a47d16a5376e1be28f368641731958004e5aa63fed"],"state_sha256":"29cb51316d9b0aa3f79e5a1bff492fbbdd6045370165bc52637d5cc92098ebae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OuclG1nCCqTnNP3STPDSoAQ/GLMF1HhefeW9QssEIYPgTMrJIX37vRfPwEUo1xie+CLGpRu1/7YKPtYSakuhDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T07:35:17.417507Z","bundle_sha256":"1fe98311e6680a9ce53514ebc020b08b88e761328a43f2e3eb5d5743f56073f5"}}