{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:HWZ3XIAJUF2AGLZMJFOTP4Q7F5","short_pith_number":"pith:HWZ3XIAJ","schema_version":"1.0","canonical_sha256":"3db3bba009a174032f2c495d37f21f2f50d3c8acf20d11323dd4c9f2f3c01312","source":{"kind":"arxiv","id":"1505.04646","version":4},"attestation_state":"computed","paper":{"title":"Proper connection number of random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ran Gu, Xueliang Li, Zhongmei Qin","submitted_at":"2015-05-18T14:07:31Z","abstract_excerpt":"A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one proper path in $G$. In this paper, we show that almost all graphs have the proper connection number 2. More precisely, let $G(n,p)$ denote the Erd\\\"{o}s-R\\'{e}nyi random graph model, in which each of the $\\binom{n}{2}$ pairs of vertices appears as an edge with probabi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1505.04646","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-18T14:07:31Z","cross_cats_sorted":[],"title_canon_sha256":"db3cc8dc8886d019c2fc26fe87966c4d0c37d44c03e1944f1109ba28274c16c4","abstract_canon_sha256":"640ccb987b0b8b9f081f6b9c22283fe366baea2cf8b69c117a0f0716002a1846"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:40:58.114951Z","signature_b64":"zgRhvgzjEGFwG/PzTrjwPj/mv/tNP9zzWPtJ6FPrdFZrV2jwW/jfDIPNOMA2cfl9C4CUqUMa1YrgfyEyR75CAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3db3bba009a174032f2c495d37f21f2f50d3c8acf20d11323dd4c9f2f3c01312","last_reissued_at":"2026-05-18T01:40:58.114223Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:40:58.114223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proper connection number of random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ran Gu, Xueliang Li, Zhongmei Qin","submitted_at":"2015-05-18T14:07:31Z","abstract_excerpt":"A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one proper path in $G$. In this paper, we show that almost all graphs have the proper connection number 2. More precisely, let $G(n,p)$ denote the Erd\\\"{o}s-R\\'{e}nyi random graph model, in which each of the $\\binom{n}{2}$ pairs of vertices appears as an edge with probabi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04646","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1505.04646","created_at":"2026-05-18T01:40:58.114341+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.04646v4","created_at":"2026-05-18T01:40:58.114341+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.04646","created_at":"2026-05-18T01:40:58.114341+00:00"},{"alias_kind":"pith_short_12","alias_value":"HWZ3XIAJUF2A","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"HWZ3XIAJUF2AGLZM","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"HWZ3XIAJ","created_at":"2026-05-18T12:29:25.134429+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HWZ3XIAJUF2AGLZMJFOTP4Q7F5","json":"https://pith.science/pith/HWZ3XIAJUF2AGLZMJFOTP4Q7F5.json","graph_json":"https://pith.science/api/pith-number/HWZ3XIAJUF2AGLZMJFOTP4Q7F5/graph.json","events_json":"https://pith.science/api/pith-number/HWZ3XIAJUF2AGLZMJFOTP4Q7F5/events.json","paper":"https://pith.science/paper/HWZ3XIAJ"},"agent_actions":{"view_html":"https://pith.science/pith/HWZ3XIAJUF2AGLZMJFOTP4Q7F5","download_json":"https://pith.science/pith/HWZ3XIAJUF2AGLZMJFOTP4Q7F5.json","view_paper":"https://pith.science/paper/HWZ3XIAJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.04646&json=true","fetch_graph":"https://pith.science/api/pith-number/HWZ3XIAJUF2AGLZMJFOTP4Q7F5/graph.json","fetch_events":"https://pith.science/api/pith-number/HWZ3XIAJUF2AGLZMJFOTP4Q7F5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HWZ3XIAJUF2AGLZMJFOTP4Q7F5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HWZ3XIAJUF2AGLZMJFOTP4Q7F5/action/storage_attestation","attest_author":"https://pith.science/pith/HWZ3XIAJUF2AGLZMJFOTP4Q7F5/action/author_attestation","sign_citation":"https://pith.science/pith/HWZ3XIAJUF2AGLZMJFOTP4Q7F5/action/citation_signature","submit_replication":"https://pith.science/pith/HWZ3XIAJUF2AGLZMJFOTP4Q7F5/action/replication_record"}},"created_at":"2026-05-18T01:40:58.114341+00:00","updated_at":"2026-05-18T01:40:58.114341+00:00"}