{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:UBVTA67FW3424NCNER5CAQLFJ4","short_pith_number":"pith:UBVTA67F","schema_version":"1.0","canonical_sha256":"a06b307be5b6f9ae344d247a2041654f1e22964e84e22db6bb1adde9403581a4","source":{"kind":"arxiv","id":"1606.06547","version":1},"attestation_state":"computed","paper":{"title":"Distance proper connection of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Colton Magnant, Meiqin Wei, Xiaoyu Zhu, Xueliang Li","submitted_at":"2016-06-21T12:53:02Z","abstract_excerpt":"Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\\ell$-proper path if no two edges of the same color appear with fewer than $\\ell$ edges in between on $P$. The graph $G$ is called $(k,\\ell)$-proper connected if every pair of distinct vertices of $G$ are connected by $k$ pairwise internally vertex-disjoint distance $\\ell$-proper paths in $G$. For a $k$-connected graph $G$, the minimum number of colors needed to make $G$ $(k,\\ell)$-proper connected is called the $(k,\\ell)$-proper connection number of $G$ and denoted by $pc_{k,\\ell}(G)$. In this paper, we prove "},"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":"1606.06547","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-21T12:53:02Z","cross_cats_sorted":[],"title_canon_sha256":"6c5ffda9ae926a952612ef1a9406bb3a887dca03af4538857d0e1766b70c30f8","abstract_canon_sha256":"c5431f60fef363d6cd68f2484cf79817b456ac8f9bd66a59455c46115b0154ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:09.279395Z","signature_b64":"vCGOSpjrOlgdpx0lQ4FvA8E9gDjFks+8GK42cRRpFi5rJUmmGbdNRRuBMRTrR6tAMEv2FraBjlRhfThxCiLtCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a06b307be5b6f9ae344d247a2041654f1e22964e84e22db6bb1adde9403581a4","last_reissued_at":"2026-05-18T01:12:09.278985Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:09.278985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distance proper connection of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Colton Magnant, Meiqin Wei, Xiaoyu Zhu, Xueliang Li","submitted_at":"2016-06-21T12:53:02Z","abstract_excerpt":"Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\\ell$-proper path if no two edges of the same color appear with fewer than $\\ell$ edges in between on $P$. The graph $G$ is called $(k,\\ell)$-proper connected if every pair of distinct vertices of $G$ are connected by $k$ pairwise internally vertex-disjoint distance $\\ell$-proper paths in $G$. For a $k$-connected graph $G$, the minimum number of colors needed to make $G$ $(k,\\ell)$-proper connected is called the $(k,\\ell)$-proper connection number of $G$ and denoted by $pc_{k,\\ell}(G)$. In this paper, we prove "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.06547","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.06547","created_at":"2026-05-18T01:12:09.279063+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.06547v1","created_at":"2026-05-18T01:12:09.279063+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.06547","created_at":"2026-05-18T01:12:09.279063+00:00"},{"alias_kind":"pith_short_12","alias_value":"UBVTA67FW342","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"UBVTA67FW3424NCN","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"UBVTA67F","created_at":"2026-05-18T12:30:46.583412+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/UBVTA67FW3424NCNER5CAQLFJ4","json":"https://pith.science/pith/UBVTA67FW3424NCNER5CAQLFJ4.json","graph_json":"https://pith.science/api/pith-number/UBVTA67FW3424NCNER5CAQLFJ4/graph.json","events_json":"https://pith.science/api/pith-number/UBVTA67FW3424NCNER5CAQLFJ4/events.json","paper":"https://pith.science/paper/UBVTA67F"},"agent_actions":{"view_html":"https://pith.science/pith/UBVTA67FW3424NCNER5CAQLFJ4","download_json":"https://pith.science/pith/UBVTA67FW3424NCNER5CAQLFJ4.json","view_paper":"https://pith.science/paper/UBVTA67F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.06547&json=true","fetch_graph":"https://pith.science/api/pith-number/UBVTA67FW3424NCNER5CAQLFJ4/graph.json","fetch_events":"https://pith.science/api/pith-number/UBVTA67FW3424NCNER5CAQLFJ4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UBVTA67FW3424NCNER5CAQLFJ4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UBVTA67FW3424NCNER5CAQLFJ4/action/storage_attestation","attest_author":"https://pith.science/pith/UBVTA67FW3424NCNER5CAQLFJ4/action/author_attestation","sign_citation":"https://pith.science/pith/UBVTA67FW3424NCNER5CAQLFJ4/action/citation_signature","submit_replication":"https://pith.science/pith/UBVTA67FW3424NCNER5CAQLFJ4/action/replication_record"}},"created_at":"2026-05-18T01:12:09.279063+00:00","updated_at":"2026-05-18T01:12:09.279063+00:00"}