{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:APW3F3O5BLPQKTK22UGVJEURQP","short_pith_number":"pith:APW3F3O5","canonical_record":{"source":{"id":"2605.27111","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T14:51:13Z","cross_cats_sorted":[],"title_canon_sha256":"9c3946e70cabe427d1ce007052d496ee85b6ac3e346c0069703b64c8a5e91802","abstract_canon_sha256":"1e94b66f8aabbe14a70e251113763df01ca397cb5f07a4602a3fe94153c91e8b"},"schema_version":"1.0"},"canonical_sha256":"03edb2eddd0adf054d5ad50d54929183da35e088d6b54636fd57df7a8f0ea8b5","source":{"kind":"arxiv","id":"2605.27111","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.27111","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"arxiv_version","alias_value":"2605.27111v1","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.27111","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"pith_short_12","alias_value":"APW3F3O5BLPQ","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"pith_short_16","alias_value":"APW3F3O5BLPQKTK2","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"pith_short_8","alias_value":"APW3F3O5","created_at":"2026-05-27T02:05:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:APW3F3O5BLPQKTK22UGVJEURQP","target":"record","payload":{"canonical_record":{"source":{"id":"2605.27111","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T14:51:13Z","cross_cats_sorted":[],"title_canon_sha256":"9c3946e70cabe427d1ce007052d496ee85b6ac3e346c0069703b64c8a5e91802","abstract_canon_sha256":"1e94b66f8aabbe14a70e251113763df01ca397cb5f07a4602a3fe94153c91e8b"},"schema_version":"1.0"},"canonical_sha256":"03edb2eddd0adf054d5ad50d54929183da35e088d6b54636fd57df7a8f0ea8b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-27T02:05:42.514929Z","signature_b64":"r7bUJ9CRQ10CkHKfFDrLh48Xn7P0gaaltzI1zd/4nGvjkgauuQCt79M84PNX3sJiAB7o3m1YW4Zx7F1e/0cOBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03edb2eddd0adf054d5ad50d54929183da35e088d6b54636fd57df7a8f0ea8b5","last_reissued_at":"2026-05-27T02:05:42.513919Z","signature_status":"signed_v1","first_computed_at":"2026-05-27T02:05:42.513919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.27111","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-27T02:05:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ywzqQGXCPNZgPgc6Tld3dSOMvmo1NCpOwFm10itYPjUqi9vR6Im7zI2ob8gpw3c0kwSJLE6o+f5JsxDEM7gJDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T15:17:30.799461Z"},"content_sha256":"ce00adebc8d71e93919390104693a14f3c0972e217c6e5889698cf0862cf850e","schema_version":"1.0","event_id":"sha256:ce00adebc8d71e93919390104693a14f3c0972e217c6e5889698cf0862cf850e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:APW3F3O5BLPQKTK22UGVJEURQP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The list r-hued coloring of trees and unicyclic graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fengxia Liu, Yu Miao","submitted_at":"2026-05-26T14:51:13Z","abstract_excerpt":"Let $r$ be a positive integer and $G$ be a graph. The list $r$-hued chromatic number of $G$, denoted by $\\chi_{L,r}(G)$, is the smallest integer $k$, such that for each $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. It is proved in [Discrete Math. 306 (16) (2006) 1997-2004] that every tree $G$ satisfies $\\chi_{r}(G)=\\min\\{r,\\Delta(G)\\}+1$. It is known that every cycle graph $C_{n}$ with order $n$ has $\\chi_{L,r}(C_{n})=\\chi_{r}(C_{n})$. The main results are the following:\n  $(1)$ If $G$ is a tree, then $\\chi_{L,r}(G)=\\min\\{r,\\Delta(G)\\}+1$;\n  $(2)$ Let $G$ be a unicyclic graph which is not "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.27111","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.27111/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-05-27T02:05:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6w/n/p5n0yKiikHHg3KUOfuqmhP4Sct0rrX7sJ2VwbNoqvwlkxukY/8SXMxbJltV7chkRXg31T8ZBVcTo7b5DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T15:17:30.800095Z"},"content_sha256":"57a1902338f0a490b146f7df9348cb9d9c4d057e12a00ea21b07266bea4212bb","schema_version":"1.0","event_id":"sha256:57a1902338f0a490b146f7df9348cb9d9c4d057e12a00ea21b07266bea4212bb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP/bundle.json","state_url":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/APW3F3O5BLPQKTK22UGVJEURQP/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":"2026-08-03T15:17:30Z","links":{"resolver":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP","bundle":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP/bundle.json","state":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/APW3F3O5BLPQKTK22UGVJEURQP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:APW3F3O5BLPQKTK22UGVJEURQP","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":"1e94b66f8aabbe14a70e251113763df01ca397cb5f07a4602a3fe94153c91e8b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T14:51:13Z","title_canon_sha256":"9c3946e70cabe427d1ce007052d496ee85b6ac3e346c0069703b64c8a5e91802"},"schema_version":"1.0","source":{"id":"2605.27111","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.27111","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"arxiv_version","alias_value":"2605.27111v1","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.27111","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"pith_short_12","alias_value":"APW3F3O5BLPQ","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"pith_short_16","alias_value":"APW3F3O5BLPQKTK2","created_at":"2026-05-27T02:05:42Z"},{"alias_kind":"pith_short_8","alias_value":"APW3F3O5","created_at":"2026-05-27T02:05:42Z"}],"graph_snapshots":[{"event_id":"sha256:57a1902338f0a490b146f7df9348cb9d9c4d057e12a00ea21b07266bea4212bb","target":"graph","created_at":"2026-05-27T02:05:42Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.27111/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $r$ be a positive integer and $G$ be a graph. The list $r$-hued chromatic number of $G$, denoted by $\\chi_{L,r}(G)$, is the smallest integer $k$, such that for each $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. It is proved in [Discrete Math. 306 (16) (2006) 1997-2004] that every tree $G$ satisfies $\\chi_{r}(G)=\\min\\{r,\\Delta(G)\\}+1$. It is known that every cycle graph $C_{n}$ with order $n$ has $\\chi_{L,r}(C_{n})=\\chi_{r}(C_{n})$. The main results are the following:\n  $(1)$ If $G$ is a tree, then $\\chi_{L,r}(G)=\\min\\{r,\\Delta(G)\\}+1$;\n  $(2)$ Let $G$ be a unicyclic graph which is not ","authors_text":"Fengxia Liu, Yu Miao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T14:51:13Z","title":"The list r-hued coloring of trees and unicyclic graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.27111","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:ce00adebc8d71e93919390104693a14f3c0972e217c6e5889698cf0862cf850e","target":"record","created_at":"2026-05-27T02:05:42Z","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":"1e94b66f8aabbe14a70e251113763df01ca397cb5f07a4602a3fe94153c91e8b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-26T14:51:13Z","title_canon_sha256":"9c3946e70cabe427d1ce007052d496ee85b6ac3e346c0069703b64c8a5e91802"},"schema_version":"1.0","source":{"id":"2605.27111","kind":"arxiv","version":1}},"canonical_sha256":"03edb2eddd0adf054d5ad50d54929183da35e088d6b54636fd57df7a8f0ea8b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"03edb2eddd0adf054d5ad50d54929183da35e088d6b54636fd57df7a8f0ea8b5","first_computed_at":"2026-05-27T02:05:42.513919Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-27T02:05:42.513919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r7bUJ9CRQ10CkHKfFDrLh48Xn7P0gaaltzI1zd/4nGvjkgauuQCt79M84PNX3sJiAB7o3m1YW4Zx7F1e/0cOBw==","signature_status":"signed_v1","signed_at":"2026-05-27T02:05:42.514929Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.27111","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce00adebc8d71e93919390104693a14f3c0972e217c6e5889698cf0862cf850e","sha256:57a1902338f0a490b146f7df9348cb9d9c4d057e12a00ea21b07266bea4212bb"],"state_sha256":"92e120db6b86c0ee2300a0127f4c23977cf2c48fdcc293598b08fa038a3e2825"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"16NMqfQv65RnZAZfoTD5ai6Mcj7msnBhw38HaNOSdAukuAdp0sX/rZgoX51N4FnL0/DeqhcjPYoGzMt12SaJCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T15:17:30.810043Z","bundle_sha256":"c14d6fd5fe633706f63c2b6c0c0240194f6334d40d617b5f82bb2a9a0566e12b"}}