{"schema_version":"dataset-v1","id":"petersen-witness-graph","version":1,"title":"The 112-vertex counterexample","description":"The main witness: a bridgeless cubic graph on 112 vertices and 168 edges with no Petersen coloring, drawn from its exact published adjacency list.","license":"CC0-1.0 (layout); adjacency as published by the cited source","problems":["petersen-coloring"],"statements":[{"problem":"petersen-coloring","statement":"petersen-112-main-witness","role":"This graph has no Petersen coloring and no normal five-edge-coloring."},{"problem":"petersen-coloring","statement":"petersen-universal-coloring","role":"So not every bridgeless cubic graph is Petersen-colorable."}],"figure":{"family":"graph","title":"The 112-vertex counterexample","caption":"Every vertex has degree three and no single edge disconnects the graph, yet no map to the Petersen graph preserves its edge-stars.","panels":[{"key":"main","label":"Main witness G = 3L + C","highlight":null}],"node_labels":false,"boundary":"Positions come from a computed layout and carry no meaning; only the edges do. The picture cannot show that no Petersen coloring exists: that is the cited, formally checked theorem.","statement":"petersen-112-main-witness"},"provenance":{"method":"Edges read from the published witness and checked against its SHA-256; layout from the Laplacian's second and third eigenvectors refined by 400 deterministic Fruchterman–Reingold steps.","software":{"python":"3.12.15","numpy":"2.3.4"},"precision":"Exact edges; positions rounded to four decimals.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"nodes","label":"vertex positions (layout only)","x":[0.0913,-0.0189,-0.2013,0.2636,0.2789,0.126,0.1578,0.0414,0.5815,0.5348,0.7131,0.4103,0.8599,0.8777,0.7501,0.8208,0.9135,0.911,0.9165,0.8754,0.6289,0.5613,0.6428,0.4302,0.233,0.5278,0.3999,0.4571,0.3282,0.2886,0.3311,0.1545,0.1889,0.2415,0.1936,0.2401,-0.0048,0.1639,-0.1497,-0.0557,0.1108,0.2124,0.2193,0.301,0.4454,0.2425,0.3249,0.3042,0.4865,0.6172,0.571,0.2825,0.7366,0.8379,0.7828,0.6772,0.8942,0.7883,0.8304,0.8632,0.4129,0.3361,0.458,0.0872,0.0579,0.2505,0.1526,0.255,0.0576,-0.0926,-0.0723,-0.0642,-0.416,-0.1038,-0.235,-0.2315,-0.7159,-0.575,-0.363,-0.8627,-0.7401,-0.7277,-0.7322,-0.5843,-0.9945,-0.8633,-0.9732,-1,-0.8496,-0.7317,-0.7517,-0.6747,-0.4078,-0.7148,-0.5789,-0.6035,-0.9203,-0.8224,-0.7378,-0.7927,-0.4853,-0.801,-0.6821,-0.6692,-0.9989,-0.8904,-0.8352,-0.8357,-0.6491,-0.9284,-0.8015,-0.7908],"y":[-0.8272,-0.8822,-0.8028,-0.9493,-0.8461,-0.9428,-0.9227,-0.8698,-0.7975,-0.7338,-0.6716,-0.8564,-0.5855,-0.4837,-0.5596,-0.377,-0.1217,-0.4603,-0.3109,-0.3609,-0.5356,-0.4085,-0.355,-0.4419,-0.2175,-0.4439,-0.335,-0.3138,-0.7193,-0.5878,-0.4992,-0.6566,-0.2306,-0.6048,-0.4715,-0.4284,0.0474,0.0834,0.1072,-0.0775,0.792,0.6439,0.4061,0.8755,0.7306,0.7831,0.7682,0.6273,0.8458,0.7359,0.7122,0.9447,0.623,0.5269,0.5901,0.4434,0.1781,0.5021,0.3663,0.4066,0.6316,0.5287,0.4639,0.6477,0.3211,0.5877,0.4953,0.4399,0.9673,0.9077,0.7837,0.8672,0.7173,0.9416,0.8501,0.8185,0.2245,0.116,0.02,0.1074,0.0034,0.1334,0.09,0.0499,-0.111,-0.0855,-0.2854,0.0623,-0.2868,-0.234,-0.1519,-0.3678,-0.0716,-0.2799,-0.2228,-0.1608,-0.437,-0.5252,-0.5149,-0.2647,-0.6624,-0.4496,-0.4951,-0.5856,0.2103,0.2355,0.0508,0.3317,0.5152,0.313,0.4045,0.3037]}],"marks":[],"edges":[[0,1],[0,5],[0,31],[1,2],[1,6],[2,7],[2,100],[3,5],[3,6],[3,11],[4,6],[4,7],[4,9],[5,7],[8,9],[8,10],[8,11],[9,14],[10,12],[10,20],[11,28],[12,13],[12,17],[13,14],[13,18],[14,19],[15,17],[15,18],[15,22],[16,18],[16,19],[16,56],[17,19],[20,21],[20,25],[21,22],[21,26],[22,27],[23,25],[23,26],[23,30],[24,26],[24,27],[24,39],[25,27],[28,29],[28,33],[29,30],[29,34],[30,35],[31,33],[31,34],[32,34],[32,35],[32,37],[33,35],[36,37],[36,38],[36,39],[37,42],[38,64],[38,92],[39,78],[40,41],[40,45],[40,71],[41,42],[41,46],[42,47],[43,45],[43,46],[43,51],[44,46],[44,47],[44,49],[45,47],[48,49],[48,50],[48,51],[49,54],[50,52],[50,60],[51,68],[52,53],[52,57],[53,54],[53,58],[54,59],[55,57],[55,58],[55,62],[56,58],[56,59],[57,59],[60,61],[60,65],[61,62],[61,66],[62,67],[63,65],[63,66],[63,70],[64,66],[64,67],[65,67],[68,69],[68,73],[69,70],[69,74],[70,75],[71,73],[71,74],[72,74],[72,75],[72,108],[73,75],[76,77],[76,81],[76,107],[77,78],[77,82],[78,83],[79,81],[79,82],[79,87],[80,82],[80,83],[80,85],[81,83],[84,85],[84,86],[84,87],[85,90],[86,88],[86,96],[87,104],[88,89],[88,93],[89,90],[89,94],[90,95],[91,93],[91,94],[91,98],[92,94],[92,95],[93,95],[96,97],[96,101],[97,98],[97,102],[98,103],[99,101],[99,102],[99,106],[100,102],[100,103],[101,103],[104,105],[104,109],[105,106],[105,110],[106,111],[107,109],[107,110],[108,110],[108,111],[109,111]]},"sha256":"bfaaec91ea6b38336952587a046456d385768250b2a695debddc4259fcb7b345"}