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If such an edge labeling induces a vertex labeling in which every vertex has a constant vertex sum (mod p), then G is said to be k-edge magic (k-EM). In this paper, we (i) show that all the maximal outerplanar graphs of order p = 4; 5; 7 are k-EM if and only if k\\equiv 2 (mod p); (ii) obtain all the maximal outerplanar graphs that are"},"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":"1207.2977","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-12T14:16:13Z","cross_cats_sorted":[],"title_canon_sha256":"f0f144cb8e70550c732d35467fa4ac0955a96c87ce129aa64e8cf173cc2bd882","abstract_canon_sha256":"045fc2cf3469b8f5cf2e4cd6e5223ef9d60d9e8bd685b61db5db98bf0a2e98d4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:12.471546Z","signature_b64":"NTBGfRl55ggdHVcvvw9zrsFB/QsUkLUwUXznmcO91w8SBRYpEJz2tx6+JRczMbvwarQyu+OU1KV4NyPPJt9fCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e72e23761b9a30d25a1933913a183220531447984853769c71b856190f96a71d","last_reissued_at":"2026-05-18T03:51:12.471045Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:12.471045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the k-edge magic graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gee-Choon Lau, Saeid Alikhani, Sin-Min Lee, William Kocay","submitted_at":"2012-07-12T14:16:13Z","abstract_excerpt":"Let $G$ be a graph with vertex set V and edge set E such that |V| = p and |E| = q. 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In this paper, we (i) show that all the maximal outerplanar graphs of order p = 4; 5; 7 are k-EM if and only if k\\equiv 2 (mod p); (ii) obtain all the maximal outerplanar graphs that are"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2977","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1207.2977","created_at":"2026-05-18T03:51:12.471106+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.2977v2","created_at":"2026-05-18T03:51:12.471106+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2977","created_at":"2026-05-18T03:51:12.471106+00:00"},{"alias_kind":"pith_short_12","alias_value":"44XCG5Q3TIYN","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"44XCG5Q3TIYNEWQZ","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"44XCG5Q3","created_at":"2026-05-18T12:26:53.410803+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/44XCG5Q3TIYNEWQZGOITUGBSEB","json":"https://pith.science/pith/44XCG5Q3TIYNEWQZGOITUGBSEB.json","graph_json":"https://pith.science/api/pith-number/44XCG5Q3TIYNEWQZGOITUGBSEB/graph.json","events_json":"https://pith.science/api/pith-number/44XCG5Q3TIYNEWQZGOITUGBSEB/events.json","paper":"https://pith.science/paper/44XCG5Q3"},"agent_actions":{"view_html":"https://pith.science/pith/44XCG5Q3TIYNEWQZGOITUGBSEB","download_json":"https://pith.science/pith/44XCG5Q3TIYNEWQZGOITUGBSEB.json","view_paper":"https://pith.science/paper/44XCG5Q3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.2977&json=true","fetch_graph":"https://pith.science/api/pith-number/44XCG5Q3TIYNEWQZGOITUGBSEB/graph.json","fetch_events":"https://pith.science/api/pith-number/44XCG5Q3TIYNEWQZGOITUGBSEB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/44XCG5Q3TIYNEWQZGOITUGBSEB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/44XCG5Q3TIYNEWQZGOITUGBSEB/action/storage_attestation","attest_author":"https://pith.science/pith/44XCG5Q3TIYNEWQZGOITUGBSEB/action/author_attestation","sign_citation":"https://pith.science/pith/44XCG5Q3TIYNEWQZGOITUGBSEB/action/citation_signature","submit_replication":"https://pith.science/pith/44XCG5Q3TIYNEWQZGOITUGBSEB/action/replication_record"}},"created_at":"2026-05-18T03:51:12.471106+00:00","updated_at":"2026-05-18T03:51:12.471106+00:00"}