{"paper":{"title":"Regular and biregular planar cages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fidel Barrera-Cruz, Gabriela Araujo-Pardo, Natalia Garc\\'ia-Col\\'in","submitted_at":"2018-11-19T01:38:34Z","abstract_excerpt":"We study the Cage Problem for regular and biregular planar graphs. A $(k,g)$-graph is a $k$-regular graph with girth $g$. A $(k,g)$-cage is a $(k,g)$-graph of minimum order. It is not difficult to conclude that the regular planar cages are the Platonic Solids. A $(\\{r,m\\};g)$-graph is a graph of girth $g$ whose vertices have degrees $r$ and $m.$ A $(\\{r,m\\};g)$-cage is a $(\\{r,m\\};g)$-graph of minimum order. In this case we determine the triplets of values $(\\{r,m\\};g)$ for which there exist planar $(\\{r,m\\};g)$--graphs, for all those values we construct examples. Furthermore, for many triplet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.07449","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"}