{"paper":{"title":"List colorings with distinct list sizes, the case of complete bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ida Kantor, Zolt\\'an F\\\"uredi","submitted_at":"2011-11-01T16:13:31Z","abstract_excerpt":"Let $f:V \\rightarrow \\mathbb{N}$ be a function on the vertex set of the graph $G=(V,E)$. The graph $G$ is {\\em $f$-choosable} if for every collection of lists with list sizes specified by $f$ there is a proper coloring using colors from the lists. The sum choice number, $\\chi_{sc}(G)$, is the minimum of $\\sum f(v)$, over all functions $f$ such that $G$ is $f$-choosable. It is known (Alon 1993, 2000) that if $G$ has average degree $d$, then the usual choice number $\\chi_\\ell(G)$ is at least $\\Omega(\\log d)$, so they grow simultaneously.\n  In this paper we show that $\\chi_{sc}(G)/|V(G)|$ can be "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.0234","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":""},"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"}