{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:53UOY3A4JCQE6ESPHWUXVJY3IL","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":"1043b308df26b5fd4d2e97029764743ab4e07f21cb10ca0bbca12432f0b2086a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-11-03T19:01:11Z","title_canon_sha256":"eba2b592f8367ca320df989cb3939029ae756390dc5e1daad9606ee4d99bec58"},"schema_version":"1.0","source":{"id":"1011.0971","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.0971","created_at":"2026-05-18T02:58:33Z"},{"alias_kind":"arxiv_version","alias_value":"1011.0971v2","created_at":"2026-05-18T02:58:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.0971","created_at":"2026-05-18T02:58:33Z"},{"alias_kind":"pith_short_12","alias_value":"53UOY3A4JCQE","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"53UOY3A4JCQE6ESP","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"53UOY3A4","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:6e35697fc41486001f9666529b84e83fb963a5886fc7ccb4df8b54198681dc3b","target":"graph","created_at":"2026-05-18T02:58:33Z","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":"The {\\em packing chromatic number} $\\chi_{\\rho}(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\\{1,2,\\ldots ,k\\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This paper studies the packing chromatic number of infinite distance graphs $G(\\mathbb{Z},D)$, i.e. graphs with the set $\\mathbb{Z}$ of integers as vertex set, with two distinct vertices $i,j\\in \\mathbb{Z}$ being adjacent if and only if $|i-j|\\in D$. We present lower and upper bounds for $\\chi_{\\rho}(G(\\mathbb{Z},D))$, showing that for finite $D$, the packing ch","authors_text":"Olivier Togni (Le2i)","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-11-03T19:01:11Z","title":"On Packing Colorings of Distance Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0971","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":"sha256:208f3107f86615393f8d06864c5d76cd587efaaec2687bae364959f8a9f90b96","target":"record","created_at":"2026-05-18T02:58:33Z","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":"1043b308df26b5fd4d2e97029764743ab4e07f21cb10ca0bbca12432f0b2086a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-11-03T19:01:11Z","title_canon_sha256":"eba2b592f8367ca320df989cb3939029ae756390dc5e1daad9606ee4d99bec58"},"schema_version":"1.0","source":{"id":"1011.0971","kind":"arxiv","version":2}},"canonical_sha256":"eee8ec6c1c48a04f124f3da97aa71b42f66a1e6632ae62e8a5d798b2d8da79cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eee8ec6c1c48a04f124f3da97aa71b42f66a1e6632ae62e8a5d798b2d8da79cd","first_computed_at":"2026-05-18T02:58:33.025223Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:33.025223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BNSQYswCsbOpo/u73DVf/sCwVKKXnJyiGldIP7bLO0m4h2M3LQFjIAxAtMFBUogDQ8hvKgPswrH6J6JpExy1BA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:33.025940Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.0971","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:208f3107f86615393f8d06864c5d76cd587efaaec2687bae364959f8a9f90b96","sha256:6e35697fc41486001f9666529b84e83fb963a5886fc7ccb4df8b54198681dc3b"],"state_sha256":"231e180e28db8b46a2974bb70b965f70c4f4dd4db114c91be8cbab9d5bed8e81"}