{"schema_version":"dataset-v1","id":"caccetta-haggkvist-regions","version":1,"title":"Where the cycle bound is proved","description":"Horizontal: the minimum out-degree r. Vertical: k = n/r, so the conjecture asks for a directed cycle of length at most ⌈k⌉. Shaded regions are cases with a cited proof: r = 2–5 for every n; n ≥ 2r² − 3r + 1, which is k ≥ 2r − 3 + 1/r; and the triangle case up to k = 1/0.3465. The open region is what remains for r ≥ 6.","license":"CC0-1.0 (exact values)","problems":["caccetta-haggkvist"],"statements":[{"problem":"caccetta-haggkvist","statement":"ch-cycle-bound","role":"The open part: r ≥ 6 between the triangle case and Shen’s curve."},{"problem":"caccetta-haggkvist","statement":"ch-small-outdegree","role":"Proved for r = 2, 3, 4, 5 and every n."},{"problem":"caccetta-haggkvist","statement":"ch-large-order","role":"Proved above the curve k = 2r − 3 + 1/r."},{"problem":"caccetta-haggkvist","statement":"ch-triangle-03465","role":"Proved for k ≤ 1/0.3465 ≈ 2.886 (oriented graphs)."},{"problem":"caccetta-haggkvist","statement":"ch-triangle-target","role":"The line k = 3, the one-third triangle question, is open."},{"problem":"caccetta-haggkvist","statement":"ch-vertex-transitive","role":"Vertex-transitive graphs: proved for every r and n."}],"figure":{"family":"region-plot","title":"Where the cycle bound is proved","caption":"Each point (r, k) stands for digraphs with minimum out-degree r on n = kr vertices. For each r ≥ 6, the open part runs from the triangle case up to Shen’s curve, which rises with r. The triangle results are stated for oriented graphs; a digraph with opposite arcs already has a cycle of length two. Vertex-transitive digraphs, a class rather than a region, satisfy the bound for every r and n.","x_label":"out-degree r","y_label":"k = n/r (cycle length ⌈k⌉)","x_range":[1.5,12.5],"y_range":[2,24],"y_scale":"log10","x_ticks":[[2,"2"],[3,"3"],[4,"4"],[5,"5"],[6,"6"],[7,"7"],[8,"8"],[9,"9"],[10,"10"],[11,"11"],[12,"12"]],"y_ticks":[[2,"2"],[3,"3"],[5,"5"],[10,"10"],[20,"20"]],"layers":[{"key":"open","label":"Open: r ≥ 6 between the triangle case and Shen’s curve","kind":"polygon","status":"open","statement":"ch-cycle-bound","x":null,"y_max":null,"series":null,"points":[[5.5,2.886002886002886],[12.5,2.886002886002886],[12.5,22.08],[12,21.083333333333332],[11.5,20.08695652173913],[11,19.09090909090909],[10.5,18.095238095238095],[10,17.1],[9.5,16.105263157894736],[9,15.11111111111111],[8.5,14.117647058823527],[8,13.125],[7.5,12.133333333333333],[7,11.142857142857142],[6.5,10.153846153846152],[6,9.166666666666666],[5.5,8.181818181818182]]},{"key":"small","label":"Proved: r = 2, 3, 4, 5 for every n","kind":"polygon","status":"proved","statement":"ch-small-outdegree","x":null,"y_max":null,"series":null,"points":[[1.5,2],[5.5,2],[5.5,24],[1.5,24]]},{"key":"large","label":"Proved: n ≥ 2r² − 3r + 1 (Shen)","kind":"polygon","status":"proved","statement":"ch-large-order","x":null,"y_max":null,"series":null,"points":[[5.5,24],[5.5,8.181818181818182],[6,9.166666666666666],[6.5,10.153846153846152],[7,11.142857142857142],[7.5,12.133333333333333],[8,13.125],[8.5,14.117647058823527],[9,15.11111111111111],[9.5,16.105263157894736],[10,17.1],[10.5,18.095238095238095],[11,19.09090909090909],[11.5,20.08695652173913],[12,21.083333333333332],[12.5,22.08],[12.5,24]]},{"key":"triangle","label":"Proved: k ≤ 2.886, the triangle case at 0.3465n","kind":"polygon","status":"proved","statement":"ch-triangle-03465","x":null,"y_max":null,"series":null,"points":[[5.5,2],[12.5,2],[12.5,2.886002886002886],[5.5,2.886002886002886]]},{"key":"third","label":"Open: k = 3, the one-third triangle question","kind":"segment","status":"open","statement":"ch-triangle-target","x":null,"y_max":null,"series":null,"points":[[5.5,3],[12.5,3]]}],"supersedes":null},"provenance":{"method":"Region boundaries from the cited statements: r ∈ {2, 3, 4, 5}; k = 2r − 3 + 1/r sampled exactly at r = 1.5, 2.0, …, 12.5 (from n ≥ 2r² − 3r + 1); k = 1/0.3465 (from minimum out-degree 0.3465n). No graphs are computed.","software":{"python":"3.12.13"},"precision":"Exact rationals, stored as floating point.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"large_order","label":"k = 2r − 3 + 1/r (n = 2r² − 3r + 1)","x":[1.5,2,2.5,3,3.5,4,4.5,5,5.5,6,6.5,7,7.5,8,8.5,9,9.5,10,10.5,11,11.5,12,12.5],"y":[0.6666666666666666,1.5,2.4,3.3333333333333335,4.285714285714286,5.25,6.222222222222222,7.2,8.181818181818182,9.166666666666666,10.153846153846152,11.142857142857142,12.133333333333333,13.125,14.117647058823527,15.11111111111111,16.105263157894736,17.1,18.095238095238095,19.09090909090909,20.08695652173913,21.083333333333332,22.08]}],"marks":[]},"sha256":"f28fbf73d57c10505363dc1cb5ccbd4cdaaf0511e401d9bf3dd655ce504f2303"}