{"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"}