{"schema_version":"dataset-v1","id":"cdc-petersen","version":1,"title":"A cycle double cover of the Petersen graph","description":"The Petersen graph is the smallest snark: bridgeless and cubic, but with no 3-edge-coloring, the case where easy constructions of a cover fail. Five cycles found by search cover each of its 15 edges exactly twice (total length 30).","license":"CC0-1.0 (computed values)","problems":["cycle-double-cover"],"statements":[{"problem":"cycle-double-cover","statement":"cycle-double-cover-existence","role":"What a cycle double cover is; this graph has one."},{"problem":"cycle-double-cover","statement":"cycle-double-cover-total-length","role":"The cycle lengths add up to 2 × 15 = 30."}],"figure":{"family":"graph","title":"A cycle double cover of the Petersen graph","caption":"Each panel highlights one cycle of the cover; every edge is highlighted in exactly two panels.","panels":[{"key":"graph","label":"The Petersen graph","highlight":null},{"key":"cycle_1","label":"Cycle 1: 0 – 1 – 2 – 3 – 4 – 0","highlight":"cycle_1"},{"key":"cycle_2","label":"Cycle 2: 0 – 1 – 6 – 8 – 5 – 0","highlight":"cycle_2"},{"key":"cycle_3","label":"Cycle 3: 0 – 4 – 9 – 7 – 5 – 0","highlight":"cycle_3"},{"key":"cycle_4","label":"Cycle 4: 2 – 3 – 8 – 6 – 9 – 7 – 2","highlight":"cycle_4"},{"key":"cycle_5","label":"Cycle 5: 1 – 2 – 7 – 5 – 8 – 3 – 4 – 9 – 6 – 1","highlight":"cycle_5"}],"node_labels":true,"boundary":"One small example, found by computer search and checked edge by edge. It is not evidence for the conjecture in general.","statement":"cycle-double-cover-total-length"},"provenance":{"method":"All cycles of the Petersen graph enumerated; a depth-first search picks five whose edge multiset uses every edge exactly twice; the cover is re-checked edge by edge.","software":{"python":"3.12.15"},"precision":"Exact combinatorics.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"nodes","label":"Petersen graph, standard drawing","x":[0,0.951057,0.587785,-0.587785,-0.951057,0,0.475528,0.293893,-0.293893,-0.475528],"y":[1,0.309017,-0.809017,-0.809017,0.309017,0.5,0.154508,-0.404508,-0.404508,0.154508]}],"marks":[],"edges":[[0,1],[1,2],[2,3],[3,4],[4,0],[0,5],[1,6],[2,7],[3,8],[4,9],[5,7],[6,8],[7,9],[8,5],[9,6]],"edge_sets":[{"key":"cycle_1","label":"Cycle 1: 0 – 1 – 2 – 3 – 4 – 0","edges":[[3,4],[2,3],[1,2],[0,1],[0,4]]},{"key":"cycle_2","label":"Cycle 2: 0 – 1 – 6 – 8 – 5 – 0","edges":[[0,1],[0,5],[6,8],[1,6],[5,8]]},{"key":"cycle_3","label":"Cycle 3: 0 – 4 – 9 – 7 – 5 – 0","edges":[[7,9],[0,4],[0,5],[4,9],[5,7]]},{"key":"cycle_4","label":"Cycle 4: 2 – 3 – 8 – 6 – 9 – 7 – 2","edges":[[6,9],[2,3],[7,9],[3,8],[6,8],[2,7]]},{"key":"cycle_5","label":"Cycle 5: 1 – 2 – 7 – 5 – 8 – 3 – 4 – 9 – 6 – 1","edges":[[3,4],[6,9],[1,2],[1,6],[3,8],[4,9],[5,8],[2,7],[5,7]]}]},"sha256":"91dc505f3b7f68185f70b341fecce38fbbb4f1452d26cba673f25f912517818d"}