{"paper":{"title":"Rooted bicubic planar maps via Dyck paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Rooted bicubic planar maps on 2n vertices correspond bijectively to Dyck paths of semilength 3n with colored ascents of length divisible by 3.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jackie N. Kaminski, Juan B. Gil","submitted_at":"2026-05-17T15:59:46Z","abstract_excerpt":"We provide a combinatorial proof of Tutte's decomposition of rooted bicubic planar maps into 3-connected components. Motivated by the framework of Bell transformations, we establish an explicit bijection between rooted bicubic planar maps on $2n$ vertices and Dyck paths of semilength $3n$ with ascents of length divisible by 3, where each $3j$-ascent is colored using one of $g_j$ colors corresponding to the rooted 3-connected bicubic maps on $2j$ vertices. Our bijection gives a constructive method for assembling all rooted bicubic planar maps from their 3-connected building blocks. We give a si"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish an explicit bijection between rooted bicubic planar maps on 2n vertices and Dyck paths of semilength 3n with ascents of length divisible by 3, where each 3j-ascent is colored using one of g_j colors corresponding to the rooted 3-connected bicubic maps on 2j vertices.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The construction assumes that the numbers g_j of rooted 3-connected bicubic maps on 2j vertices are already known or recursively available independently of the full map enumeration, so that the coloring step does not presuppose the decomposition being proved.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An explicit bijection maps rooted bicubic planar maps on 2n vertices to colored Dyck paths of semilength 3n, proving Tutte's decomposition into 3-connected components via Bell transformations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Rooted bicubic planar maps on 2n vertices correspond bijectively to Dyck paths of semilength 3n with colored ascents of length divisible by 3.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b2ca136cc28eb463cfb3c0b8c43548834601afd5115808b592a39cfbe912d134"},"source":{"id":"2605.17515","kind":"arxiv","version":1},"verdict":{"id":"31bcd3f8-ed4e-489d-9a65-f50f18163151","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:31:26.474105Z","strongest_claim":"We establish an explicit bijection between rooted bicubic planar maps on 2n vertices and Dyck paths of semilength 3n with ascents of length divisible by 3, where each 3j-ascent is colored using one of g_j colors corresponding to the rooted 3-connected bicubic maps on 2j vertices.","one_line_summary":"An explicit bijection maps rooted bicubic planar maps on 2n vertices to colored Dyck paths of semilength 3n, proving Tutte's decomposition into 3-connected components via Bell transformations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The construction assumes that the numbers g_j of rooted 3-connected bicubic maps on 2j vertices are already known or recursively available independently of the full map enumeration, so that the coloring step does not presuppose the decomposition being proved.","pith_extraction_headline":"Rooted bicubic planar maps on 2n vertices correspond bijectively to Dyck paths of semilength 3n with colored ascents of length divisible by 3."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17515/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.514974Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:41:16.776526Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.650852Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.628059Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"773eb9ba59eb00a4ed30485a2abd71dd28fe32fb56b065868d32ba0ad3b21b67"},"references":{"count":8,"sample":[{"doi":"","year":2016,"title":"J.-L. Baril, R. Genestier, A. Giorgetti, and A. Petrossian, Rooted planar maps modulo some patterns, Discrete Math.339(2016), 1199–1205","work_id":"afd5a315-01ee-47b2-a80f-f4f828757407","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"D. Birmajer, J. Gil, P. McNamara, and M. Weiner, Enumeration of colored Dyck paths via partial Bell polynomials,Lattice Path Combinatorics and Applications, G.E. Andrews, C. Krattenthaler, A. Krinik (","work_id":"4aa99979-d1db-416b-8682-9de844a114c5","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1989,"title":"Batagelj, An inductive definition of the class of 3-connected quadrangulations of the plane,Discrete Math.78(1989), 45–53","work_id":"42aa7ca3-39b7-414e-b851-fa31a21c0903","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"D. Birmajer, J. Gil, and M. Weiner, A family of Bell transformations,Discrete Math.342(2019), 38–54","work_id":"ec9014dc-8b61-4dd5-8323-04875770d3d4","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"E. Horev, M. J. Katz, R. Krakovski, A. Nakamoto, Polychromatic 4-coloring of cubic bipartite plane graphs,Discrete Math.312(2012), 715–719. ROOTED BICUBIC PLANAR MAPS VIA DYCK PATHS 17 Figure 27.Roote","work_id":"04fdf330-f95f-49ef-82d6-1e0b067c3617","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":8,"snapshot_sha256":"1d1c52ecebd693cb9836a6a6fbbb5136eb2f76a2c0023eae978d21701e72a10a","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"}