{"paper":{"title":"Well-hued graphs with first difference two","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Geoffrey Boyer, Jeremy Lyle, Kirsti Kuenzel, Ryan Pellico","submitted_at":"2025-06-05T13:02:47Z","abstract_excerpt":"A graph $G$ is said to be well-hued if every maximal $k$-colorable subgraph of $G$ has the same order $a_k$. Therefore, if $G$ is well-hued, we can associate with $G$ a sequence $\\{a_k\\}$. Necessary and sufficient conditions were given as to when a sequence $\\{a_k\\}$ is realized by a well-hued graph. Further, it was conjectured there is only one connected well-hued graph with $a_2 = a_1 + 2$ for every $a_1 \\ge 4$. In this paper, we prove this conjecture as well as characterize nearly all well-hued graphs with $a_1=2$. We also investigate when both $G$ and its complement are well-hued."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.04993","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/2506.04993/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"}