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The proper orientation number of $G$ is defined as $ \\overrightarrow{\\chi} (G) =\\displaystyle \\min_{D\\in \\Gamma} \\displaystyle\\max_{v\\in V(G)} d^{-}_{D}(v) $ where $\\Gamma$ is the set of proper orientations of $G$. We have $ \\chi(G)-1 \\leq \\overrightarrow{\\chi} (G)\\leq \\Delta(G) $. We show that, it is $ \\m"},"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":"1305.6432","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-05-28T09:44:45Z","cross_cats_sorted":["cs.DM","cs.DS","math.CO"],"title_canon_sha256":"97ddc69ec5c06ea92ed1ecfe17f1572a1d290683ac835ba9c26571797145e909","abstract_canon_sha256":"bb266b357cd5fff4e915ecd578441537f396c1f23f3a74c9a8cd43cce23a09fe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:22.969441Z","signature_b64":"Nyi9yFA/R3X/GfxzOkYLDYDsp9EkXbuWuaVDOe5CwQzDcNeSDizR6jVB5LanTiOXmJBCq0wgYbjtGFv/09uzCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e32d0a483fe4dd1a4779c75c4550ca165cd0fa68e4429fadf62177a23f1f2ee4","last_reissued_at":"2026-05-18T02:50:22.968950Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:22.968950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Complexity of the Proper Orientation Number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS","math.CO"],"primary_cat":"cs.CC","authors_text":"Ali Dehghan, Arash Ahadi","submitted_at":"2013-05-28T09:44:45Z","abstract_excerpt":"Graph orientation is a well-studied area of graph theory. A proper orientation of a graph $G = (V,E)$ is an orientation $D$ of $E(G)$ such that for every two adjacent vertices $ v $ and $ u $, $ d^{-}_{D}(v) \\neq d^{-}_{D}(u)$ where $d_{D}^{-}(v)$ is the number of edges with head $v$ in $D$. The proper orientation number of $G$ is defined as $ \\overrightarrow{\\chi} (G) =\\displaystyle \\min_{D\\in \\Gamma} \\displaystyle\\max_{v\\in V(G)} d^{-}_{D}(v) $ where $\\Gamma$ is the set of proper orientations of $G$. We have $ \\chi(G)-1 \\leq \\overrightarrow{\\chi} (G)\\leq \\Delta(G) $. We show that, it is $ \\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6432","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":"1305.6432","created_at":"2026-05-18T02:50:22.969018+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.6432v1","created_at":"2026-05-18T02:50:22.969018+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6432","created_at":"2026-05-18T02:50:22.969018+00:00"},{"alias_kind":"pith_short_12","alias_value":"4MWQUSB74TOR","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"4MWQUSB74TORUR3Z","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"4MWQUSB7","created_at":"2026-05-18T12:27:34.582898+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/4MWQUSB74TORUR3ZY5OEKUGKCZ","json":"https://pith.science/pith/4MWQUSB74TORUR3ZY5OEKUGKCZ.json","graph_json":"https://pith.science/api/pith-number/4MWQUSB74TORUR3ZY5OEKUGKCZ/graph.json","events_json":"https://pith.science/api/pith-number/4MWQUSB74TORUR3ZY5OEKUGKCZ/events.json","paper":"https://pith.science/paper/4MWQUSB7"},"agent_actions":{"view_html":"https://pith.science/pith/4MWQUSB74TORUR3ZY5OEKUGKCZ","download_json":"https://pith.science/pith/4MWQUSB74TORUR3ZY5OEKUGKCZ.json","view_paper":"https://pith.science/paper/4MWQUSB7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.6432&json=true","fetch_graph":"https://pith.science/api/pith-number/4MWQUSB74TORUR3ZY5OEKUGKCZ/graph.json","fetch_events":"https://pith.science/api/pith-number/4MWQUSB74TORUR3ZY5OEKUGKCZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4MWQUSB74TORUR3ZY5OEKUGKCZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4MWQUSB74TORUR3ZY5OEKUGKCZ/action/storage_attestation","attest_author":"https://pith.science/pith/4MWQUSB74TORUR3ZY5OEKUGKCZ/action/author_attestation","sign_citation":"https://pith.science/pith/4MWQUSB74TORUR3ZY5OEKUGKCZ/action/citation_signature","submit_replication":"https://pith.science/pith/4MWQUSB74TORUR3ZY5OEKUGKCZ/action/replication_record"}},"created_at":"2026-05-18T02:50:22.969018+00:00","updated_at":"2026-05-18T02:50:22.969018+00:00"}