[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"branding":3,"analytics":7,"article-new-proof-nails-down-self-complementary-digraphs-on-six-nodes":10,"sections":41},{"siteName":4,"siteTagline":5,"publisherName":4,"contactEmail":6},"The Revision","Tech news, decoded.","editor@therevision.news",{"gaMeasurementId":8,"adsenseClientId":9},"G-ZW2MV82GYR","ca-pub-8533917693782264",{"article":11},{"id":12,"slug":13,"title":14,"dek":15,"body_md":16,"tags_json":17,"published_at":18,"created_at":19,"updated_at":20,"status":21,"review_note":22,"review_notes":23,"image_url":22,"persona_id":22,"persona_name":22,"section":30,"tags":31,"sources":36,"feedback":40,"feedback_at":22,"cost_usd":40,"total_tokens":40},6898,"new-proof-nails-down-self-complementary-digraphs-on-six-nodes","New Proof Nails Down Self-Complementary Digraphs on Six Nodes","A new proof pins the six-vertex threshold for self-complementary digraph completions at seven arcs, then maps every way completion can just barely fail.","Mathematicians have pinned the exact point where a small directed network is guaranteed to fit inside a perfectly self-symmetric one - and where that guarantee runs out.\n\nA loopless digraph is self-complementary if swapping every arc for a non-arc, and every non-arc for an arc, produces a digraph isomorphic to the original - a definition that has nothing to do with reversing arc direction. The researchers set out to find the largest number of arcs, q, such that any six-vertex digraph with at most q arcs is guaranteed to sit inside some self-complementary digraph of the same size. The paper proves that number is exactly 7, with the upper bound demonstrated by a disjoint union of a 3-vertex complete digraph and a 3-vertex directed path. Push past seven arcs and the guarantee can fail: the authors catalog all eight-arc digraphs that resist completion, finding five distinct isomorphism classes that collapse to three once digraphs are identified with their converse - the separate operation of reversing every arc's direction, unrelated to the complement used in the main definition.\n\nSelf-complementary graphs are a decades-old staple of combinatorics, useful in decomposition and design-theory arguments, but exact small-order thresholds like this one are notoriously fiddly to nail down by hand rather than by brute computer search. The result also settles a subtler point: every one of the five obstruction digraphs still packs alongside an isomorphic copy of itself, meaning ordinary graph packing is a strictly weaker property than same-order self-complementary completion - and the two ideas already diverge at six vertices.\n\nNo product roadmap points at this one - it's pure structural math - but exact answers at the smallest failing case are exactly what later general proofs get built and tested against.","[\"graph-theory\",\"combinatorics\",\"mathematics\",\"computer-science\"]","2026-09-18T04:00:00.000Z","2026-09-18T21:39:34.144Z","2026-09-18T21:39:46.051Z","published",null,[24],{"id":25,"reviewer":26,"round":27,"reason":28,"status":29},"editor-r1","editor",1,"Fix the technical definition: a self-complementary digraph is isomorphic to its own complement, not to a 'reverse-and-complement' — as written this conflates complement with the separate 'converse' operation the piece later uses for the isomorphism-class count, so reconcile the two uses of 'reverse' before republishing.","resolved","science",[32,33,34,35],"graph-theory","combinatorics","mathematics","computer-science",[37],{"name":38,"url":39},"arXiv cs.AI","https:\u002F\u002Farxiv.org\u002Fabs\u002F2609.20231",0,{"sections":42},[43,47,51,56,61,65,68,73,77,82,87,92,97,102],{"name":44,"slug":45,"count":46,"latest_published_at":18},"AI","ai",4082,{"name":48,"slug":49,"count":50,"latest_published_at":18},"Security","security",661,{"name":52,"slug":53,"count":54,"latest_published_at":55},"Policy","policy",339,"2026-09-17T12:00:00.000Z",{"name":57,"slug":58,"count":59,"latest_published_at":60},"Deals","deals",179,"2026-06-29T20:02:07.000Z",{"name":62,"slug":63,"count":64,"latest_published_at":18},"Hardware","hardware",155,{"name":66,"slug":30,"count":67,"latest_published_at":18},"Science",125,{"name":69,"slug":70,"count":71,"latest_published_at":72},"Consumer Tech","consumer-tech",99,"2026-09-09T17:27:33.000Z",{"name":74,"slug":75,"count":76,"latest_published_at":18},"Dev Tools","dev-tools",78,{"name":78,"slug":79,"count":80,"latest_published_at":81},"Software","software",75,"2026-09-10T20:41:21.000Z",{"name":83,"slug":84,"count":85,"latest_published_at":86},"Startups","startups",55,"2026-09-09T23:14:29.000Z",{"name":88,"slug":89,"count":90,"latest_published_at":91},"Gaming","gaming",43,"2026-09-10T12:18:06.000Z",{"name":93,"slug":94,"count":95,"latest_published_at":96},"General","general",41,"2026-09-08T01:57:23.000Z",{"name":98,"slug":99,"count":100,"latest_published_at":101},"Reviews","reviews",20,"2026-06-24T12:00:01.000Z",{"name":103,"slug":104,"count":105,"latest_published_at":106},"How-To","how-to",6,"2026-06-16T09:00:00.000Z"]