{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4YSLCFZB4M4LJQ6EMQ56IOBMGC","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":"469acb7db032626bbb81f4876341c2161960e99d9109a6719ae2985da110440a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-02-26T20:11:29Z","title_canon_sha256":"ee80fc07033a2447e36ca91f34e022b6251c892d5444f1caa5ad368ab59e42c8"},"schema_version":"1.0","source":{"id":"1302.6561","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.6561","created_at":"2026-05-18T00:29:11Z"},{"alias_kind":"arxiv_version","alias_value":"1302.6561v1","created_at":"2026-05-18T00:29:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.6561","created_at":"2026-05-18T00:29:11Z"},{"alias_kind":"pith_short_12","alias_value":"4YSLCFZB4M4L","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4YSLCFZB4M4LJQ6E","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4YSLCFZB","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:4a36ca2406abd70a5bb314e3fbef958f6dd0c2afd20a68180923299a926c939f","target":"graph","created_at":"2026-05-18T00:29:11Z","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":"A $\\Gamma$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $f$ from $V$ to an Abelian group $\\Gamma$ of order $n$ such that the weight $w(x)=\\sum_{y\\in N_G(x)}f(y)$ of every vertex $x \\in V$ is equal to the same element $\\mu \\in \\Gamma$, called the \\emph{magic constant}.\n  In this paper we will show that if $G$ is a graph of order $n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that $\\deg(v)\\equiv c \\imod {2^{p+2}}$ for some constant $c$ for any $v\\in V(G)$, then there exists a $\\Gamma$-distance magic labeling for any Abelian group $\\Gamma$ of order $4n$ for ","authors_text":"Sylwia Cichacz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-02-26T20:11:29Z","title":"Group distance magic graphs $G\\times C_n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.6561","kind":"arxiv","version":1},"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":"sha256:69022d089a85c55ff3d20f87b45bc1c1187cf22eb75d1860861b9a7c604e61be","target":"record","created_at":"2026-05-18T00:29:11Z","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":"469acb7db032626bbb81f4876341c2161960e99d9109a6719ae2985da110440a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-02-26T20:11:29Z","title_canon_sha256":"ee80fc07033a2447e36ca91f34e022b6251c892d5444f1caa5ad368ab59e42c8"},"schema_version":"1.0","source":{"id":"1302.6561","kind":"arxiv","version":1}},"canonical_sha256":"e624b11721e338b4c3c4643be4382c309e0ba1d6cc5fdb6df4f5a4fe342723c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e624b11721e338b4c3c4643be4382c309e0ba1d6cc5fdb6df4f5a4fe342723c0","first_computed_at":"2026-05-18T00:29:11.185622Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:11.185622Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eM1U07e2HQexs60bIg0slyAwo9iZx0k/UwoYIE3OMiAhAJtAXoE6iCx4o3PzC9zTV4hFne9SAyJ0YvXd/m+SCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:11.186109Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.6561","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:69022d089a85c55ff3d20f87b45bc1c1187cf22eb75d1860861b9a7c604e61be","sha256:4a36ca2406abd70a5bb314e3fbef958f6dd0c2afd20a68180923299a926c939f"],"state_sha256":"c31befed6ec90df3f0dea90592890894984f107bfdbf4a5196b95d7b5ff42e45"}