{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:VPFYEGIBMXF24BABLAKQNBMZOD","short_pith_number":"pith:VPFYEGIB","canonical_record":{"source":{"id":"1502.03977","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-13T13:13:11Z","cross_cats_sorted":[],"title_canon_sha256":"c25669f3ba001f13a84575b8f2d8feab46756b1fbed90396d603c3cfb3af4386","abstract_canon_sha256":"c5273873e9e9cd97dfdff64da32b061e39084e0a1ee840966aac85366f16ea36"},"schema_version":"1.0"},"canonical_sha256":"abcb82190165cbae0401581506859970c2bf27adcc721632f13a33e7304da0a4","source":{"kind":"arxiv","id":"1502.03977","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.03977","created_at":"2026-05-18T01:25:15Z"},{"alias_kind":"arxiv_version","alias_value":"1502.03977v2","created_at":"2026-05-18T01:25:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.03977","created_at":"2026-05-18T01:25:15Z"},{"alias_kind":"pith_short_12","alias_value":"VPFYEGIBMXF2","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"VPFYEGIBMXF24BAB","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"VPFYEGIB","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:VPFYEGIBMXF24BABLAKQNBMZOD","target":"record","payload":{"canonical_record":{"source":{"id":"1502.03977","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-13T13:13:11Z","cross_cats_sorted":[],"title_canon_sha256":"c25669f3ba001f13a84575b8f2d8feab46756b1fbed90396d603c3cfb3af4386","abstract_canon_sha256":"c5273873e9e9cd97dfdff64da32b061e39084e0a1ee840966aac85366f16ea36"},"schema_version":"1.0"},"canonical_sha256":"abcb82190165cbae0401581506859970c2bf27adcc721632f13a33e7304da0a4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:15.909897Z","signature_b64":"At71VQf0eK17gSmdmy1K0rYnaZEkxKcdhYf2kDLuAyt6ZC+WE30nw3xdE7z47NI3mIDADBeaCbBQptK+4TsfCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abcb82190165cbae0401581506859970c2bf27adcc721632f13a33e7304da0a4","last_reissued_at":"2026-05-18T01:25:15.909312Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:15.909312Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.03977","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:25:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QHpf6jIZlKo7ULYFJxF0PPh83NPdRRn2kVIqCVFB4ivImdxOgOzQf1CIJfbKxXn5Ler9UkJT4zdne06HFH4jCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:17:48.234756Z"},"content_sha256":"0679adfc00fb080790674f942f3fe7136f6d678ef8d2b1490d2fdda04cbc5a8e","schema_version":"1.0","event_id":"sha256:0679adfc00fb080790674f942f3fe7136f6d678ef8d2b1490d2fdda04cbc5a8e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:VPFYEGIBMXF24BABLAKQNBMZOD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Choosability and paintability of the lexicographic product of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Keszegh, Xuding Zhu","submitted_at":"2015-02-13T13:13:11Z","abstract_excerpt":"This paper studies the choice number and paint number of the lexicographic product of graphs. We prove that if $G$ has maximum degree $\\Delta$, then for any graph $H$ on $n$ vertices $ch(G[H]) \\le (4\\Delta+2)(ch(H) +\\log_2 n)$ and $\\chi_P(G[H]) \\le (4\\Delta+2) (\\chi_P(H)+ \\log_2 n)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03977","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:25:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AmnqjGRUYJpwiVkfApWHP2xydswmyx0BvpziwKtf1tPCEw1+R35b3Y9MG8vy2XZ+wF9f62rTgv2b0EpirxtMCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:17:48.235425Z"},"content_sha256":"b7774d663f46132db148d932e37fbf71d33b43b266e2822ea69d3721846942de","schema_version":"1.0","event_id":"sha256:b7774d663f46132db148d932e37fbf71d33b43b266e2822ea69d3721846942de"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VPFYEGIBMXF24BABLAKQNBMZOD/bundle.json","state_url":"https://pith.science/pith/VPFYEGIBMXF24BABLAKQNBMZOD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VPFYEGIBMXF24BABLAKQNBMZOD/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-25T20:17:48Z","links":{"resolver":"https://pith.science/pith/VPFYEGIBMXF24BABLAKQNBMZOD","bundle":"https://pith.science/pith/VPFYEGIBMXF24BABLAKQNBMZOD/bundle.json","state":"https://pith.science/pith/VPFYEGIBMXF24BABLAKQNBMZOD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VPFYEGIBMXF24BABLAKQNBMZOD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VPFYEGIBMXF24BABLAKQNBMZOD","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":"c5273873e9e9cd97dfdff64da32b061e39084e0a1ee840966aac85366f16ea36","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-13T13:13:11Z","title_canon_sha256":"c25669f3ba001f13a84575b8f2d8feab46756b1fbed90396d603c3cfb3af4386"},"schema_version":"1.0","source":{"id":"1502.03977","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.03977","created_at":"2026-05-18T01:25:15Z"},{"alias_kind":"arxiv_version","alias_value":"1502.03977v2","created_at":"2026-05-18T01:25:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.03977","created_at":"2026-05-18T01:25:15Z"},{"alias_kind":"pith_short_12","alias_value":"VPFYEGIBMXF2","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"VPFYEGIBMXF24BAB","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"VPFYEGIB","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:b7774d663f46132db148d932e37fbf71d33b43b266e2822ea69d3721846942de","target":"graph","created_at":"2026-05-18T01:25:15Z","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":"This paper studies the choice number and paint number of the lexicographic product of graphs. We prove that if $G$ has maximum degree $\\Delta$, then for any graph $H$ on $n$ vertices $ch(G[H]) \\le (4\\Delta+2)(ch(H) +\\log_2 n)$ and $\\chi_P(G[H]) \\le (4\\Delta+2) (\\chi_P(H)+ \\log_2 n)$.","authors_text":"Bal\\'azs Keszegh, Xuding Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-13T13:13:11Z","title":"Choosability and paintability of the lexicographic product of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03977","kind":"arxiv","version":2},"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":"sha256:0679adfc00fb080790674f942f3fe7136f6d678ef8d2b1490d2fdda04cbc5a8e","target":"record","created_at":"2026-05-18T01:25:15Z","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":"c5273873e9e9cd97dfdff64da32b061e39084e0a1ee840966aac85366f16ea36","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-13T13:13:11Z","title_canon_sha256":"c25669f3ba001f13a84575b8f2d8feab46756b1fbed90396d603c3cfb3af4386"},"schema_version":"1.0","source":{"id":"1502.03977","kind":"arxiv","version":2}},"canonical_sha256":"abcb82190165cbae0401581506859970c2bf27adcc721632f13a33e7304da0a4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"abcb82190165cbae0401581506859970c2bf27adcc721632f13a33e7304da0a4","first_computed_at":"2026-05-18T01:25:15.909312Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:15.909312Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"At71VQf0eK17gSmdmy1K0rYnaZEkxKcdhYf2kDLuAyt6ZC+WE30nw3xdE7z47NI3mIDADBeaCbBQptK+4TsfCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:15.909897Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.03977","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0679adfc00fb080790674f942f3fe7136f6d678ef8d2b1490d2fdda04cbc5a8e","sha256:b7774d663f46132db148d932e37fbf71d33b43b266e2822ea69d3721846942de"],"state_sha256":"f35f8593b3207506dfdf1c642242f5ee633ec5b4ee0d39efdeda1c1a00daee67"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mbo7dsvPVmsPlYfQYdRpuX37guzYNFWmeIdNzdHdxsAa/QmYmK4hciYPFJBEDLHcDxGULNL1GwPOmEi9neHMCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T20:17:48.239250Z","bundle_sha256":"6c2eb942d363ad90b955985968757d632ff8cd44768b28dca144e30430ed6e6e"}}