{"paper":{"title":"$(1, k)$-coloring of graphs with girth at least $5$ on a surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Geewon Suh, Hojin Choi, Ilkyoo Choi, Jisu Jeong","submitted_at":"2014-12-01T04:29:37Z","abstract_excerpt":"A graph is $(d_1, ..., d_r)$-colorable if its vertex set can be partitioned into $r$ sets $V_1, ..., V_r$ so that the maximum degree of the graph induced by $V_i$ is at most $d_i$ for each $i\\in \\{1, ..., r\\}$. For a given pair $(g, d_1)$, the question of determining the minimum $d_2=d_2(g; d_1)$ such that planar graphs with girth at least $g$ are $(d_1, d_2)$-colorable has attracted much interest. The finiteness of $d_2(g; d_1)$ was known for all cases except when $(g, d_1)=(5, 1)$. Montassier and Ochem explicitly asked if $d_2(5; 1)$ is finite. We answer this question in the affirmative with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.0344","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"}