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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 "},"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":"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"},"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"},"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.27111","created_at":"2026-05-27T02:05:42.514043+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.27111v1","created_at":"2026-05-27T02:05:42.514043+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.27111","created_at":"2026-05-27T02:05:42.514043+00:00"},{"alias_kind":"pith_short_12","alias_value":"APW3F3O5BLPQ","created_at":"2026-05-27T02:05:42.514043+00:00"},{"alias_kind":"pith_short_16","alias_value":"APW3F3O5BLPQKTK2","created_at":"2026-05-27T02:05:42.514043+00:00"},{"alias_kind":"pith_short_8","alias_value":"APW3F3O5","created_at":"2026-05-27T02:05:42.514043+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/APW3F3O5BLPQKTK22UGVJEURQP","json":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP.json","graph_json":"https://pith.science/api/pith-number/APW3F3O5BLPQKTK22UGVJEURQP/graph.json","events_json":"https://pith.science/api/pith-number/APW3F3O5BLPQKTK22UGVJEURQP/events.json","paper":"https://pith.science/paper/APW3F3O5"},"agent_actions":{"view_html":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP","download_json":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP.json","view_paper":"https://pith.science/paper/APW3F3O5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.27111&json=true","fetch_graph":"https://pith.science/api/pith-number/APW3F3O5BLPQKTK22UGVJEURQP/graph.json","fetch_events":"https://pith.science/api/pith-number/APW3F3O5BLPQKTK22UGVJEURQP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP/action/storage_attestation","attest_author":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP/action/author_attestation","sign_citation":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP/action/citation_signature","submit_replication":"https://pith.science/pith/APW3F3O5BLPQKTK22UGVJEURQP/action/replication_record"}},"created_at":"2026-05-27T02:05:42.514043+00:00","updated_at":"2026-05-27T02:05:42.514043+00:00"}