{"paper":{"title":"Triangulating Almost-Complete Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kayla Wright, Kim Nguyen Pham, Landon Settle, Padraic Bartlett","submitted_at":"2016-12-13T09:16:03Z","abstract_excerpt":"A triangle decomposition of a graph $G$ is a partition of the edges of $G$ into triangles. Two necessary conditions for $G$ to admit such a decomposition are that $|E(G)|$ is a multiple of three and that the degree of any vertex in $G$ is even; we call such graphs tridivisible.\n  Kirkman's work on Steiner triple systems established that for $G \\simeq K_n$, $G$ admits a triangle decomposition precisely when $G$ is tridivisible. In 1970, Nash-Williams conjectured that tridivisiblity is also sufficient for \"almost-complete\" graphs, which for this talk's purposes we interpret as any graph $G$ on $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04069","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"}