Pair of orthogonal k-cycle decompositions of the complete graph of order v = 2nk+1. The first decomposition is given by a set B of base cycles, the second one can be obtained multiplying every cycle of B by w. ##################################################################################### n = 1. Base cycles of a cyclic 3-cycle decomposition of the complete graph of order 7: w = -1. [ [ 1, 2, 4 ] ] n = 1. Base cycles of a cyclic 4-cycle decomposition of the complete graph of order 9: w = -1. [ [ 1, 2, 6, 4 ] ] n = 2. Base cycles of a cyclic 4-cycle decomposition of the complete graph of order 17: w = -1. [ [ 0, 1, 15, 9 ], [ 0, 2, 15, 10 ] ] n = 3. Base cycles of a cyclic 4-cycle decomposition of the complete graph of order 25: w = -1. [ [ 0, 1, 22, 13 ], [ 0, 2, 22, 14 ], [ 0, 3, 22, 15 ] ] n = 3. Base cycles of a cyclic 5-cycle decomposition of the complete graph of order 31: w = -1. [ [ 0, 3, 1, 16, 6 ], [ 0, 4, 3, 25, 12 ], [ 0, 7, 2, 25, 11 ] ] n = 4. Base cycles of a cyclic 5-cycle decomposition of the complete graph of order 41: w = 2. [ [ 0, 2, 1, 33, 13 ], [ 0, 6, 3, 33, 15 ], [ 0, 7, 2, 33, 14 ], [ 0, 8, 4, 33, 16 ] ] n = 2. Base cycles of a cyclic 6-cycle decomposition of the complete graph of order 25: w = 2. [ [ 0, 1, 23, 3, 21, 11 ], [ 0, 2, 23, 4, 21, 12 ] ] n = 4. Base cycles of a cyclic 6-cycle decomposition of the complete graph of order 49: w = 2. [ [ 0, 1, 45, 5, 41, 21 ], [ 0, 2, 45, 6, 41, 22 ], [ 0, 3, 45, 7, 41, 23 ], [ 0, 4, 45, 8, 41, 24 ] ] n = 1. Base cycles of a cyclic 7-cycle decomposition of the complete graph of order 15: w = 2. [ [ 0, 12, 3, 8, 4, 5, 13 ] ] n = 4. Base cycles of a cyclic 7-cycle decomposition of the complete graph of order 57: w = 2. [ [ 0, 52, 2, 49, 26, 45, 30 ], [ 0, 53, 4, 49, 28, 45, 32 ], [ 0, 54, 3, 49, 27, 45, 31 ], [ 0, 56, 1, 49, 25, 45, 29 ] ] n = 5. Base cycles of a cyclic 7-cycle decomposition of the complete graph of order 71: w = 2. [ [ 0, 63, 1, 61, 31, 56, 36 ], [ 0, 66, 5, 61, 35, 56, 40 ], [ 0, 67, 3, 61, 33, 56, 38 ], [ 0, 69, 4, 61, 34, 56, 39 ], [ 0, 70, 2, 61, 32, 56, 37 ] ] n = 1. Base cycles of a cyclic 8-cycle decomposition of the complete graph of order 17: w = 2. [ [ 1, 15, 3, 4, 12, 6, 10, 8 ] ] n = 2. Base cycles of a cyclic 8-cycle decomposition of the complete graph of order 33: w = 2. [ [ 0, 1, 31, 3, 29, 15, 27, 17 ], [ 0, 2, 31, 4, 29, 16, 27, 18 ] ] n = 3. Base cycles of a cyclic 8-cycle decomposition of the complete graph of order 49: w = 2. [ [ 0, 1, 46, 4, 43, 22, 40, 25 ], [ 0, 2, 46, 5, 43, 23, 40, 26 ], [ 0, 3, 46, 6, 43, 24, 40, 27 ] ] n = 4. Base cycles of a cyclic 8-cycle decomposition of the complete graph of order 65: w = 2. [ [ 0, 1, 61, 5, 57, 29, 53, 33 ], [ 0, 2, 61, 6, 57, 30, 53, 34 ], [ 0, 3, 61, 7, 57, 31, 53, 35 ], [ 0, 4, 61, 8, 57, 32, 53, 36 ] ] n = 5. Base cycles of a cyclic 8-cycle decomposition of the complete graph of order 81: w = 2. [ [ 0, 1, 76, 6, 71, 36, 66, 41 ], [ 0, 2, 76, 7, 71, 37, 66, 42 ], [ 0, 3, 76, 8, 71, 38, 66, 43 ], [ 0, 4, 76, 9, 71, 39, 66, 44 ], [ 0, 5, 76, 10, 71, 40, 66, 45 ] ] n = 6. Base cycles of a cyclic 8-cycle decomposition of the complete graph of order 97: w = 2. [ [ 0, 1, 91, 7, 85, 43, 79, 49 ], [ 0, 2, 91, 8, 85, 44, 79, 50 ], [ 0, 3, 91, 9, 85, 45, 79, 51 ], [ 0, 4, 91, 10, 85, 46, 79, 52 ], [ 0, 5, 91, 11, 85, 47, 79, 53 ], [ 0, 6, 91, 12, 85, 48, 79, 54 ] ] n = 7. Base cycles of a cyclic 8-cycle decomposition of the complete graph of order 113: w = 2. [ [ 0, 1, 106, 8, 99, 50, 92, 57 ], [ 0, 2, 106, 9, 99, 51, 92, 58 ], [ 0, 3, 106, 10, 99, 52, 92, 59 ], [ 0, 4, 106, 11, 99, 53, 92, 60 ], [ 0, 5, 106, 12, 99, 54, 92, 61 ], [ 0, 6, 106, 13, 99, 55, 92, 62 ], [ 0, 7, 106, 14, 99, 56, 92, 63 ] ] n = 3. Base cycles of a cyclic 9-cycle decomposition of the complete graph of order 55: w = 3. [ [ 0, 50, 2, 49, 5, 46, 23, 43, 26 ], [ 0, 54, 3, 49, 6, 46, 24, 43, 27 ], [ 1, 53, 0, 49, 4, 46, 15, 40, 19 ] ] n = 4. Base cycles of a cyclic 9-cycle decomposition of the complete graph of order 73: w = 2. [ [ 0, 68, 2, 65, 6, 61, 30, 57, 34 ], [ 0, 69, 4, 65, 8, 61, 32, 57, 36 ], [ 0, 70, 3, 65, 7, 61, 31, 57, 35 ], [ 0, 72, 1, 65, 5, 61, 29, 57, 33 ] ] n = 7. Base cycles of a cyclic 9-cycle decomposition of the complete graph of order 127: w = -1. [ [ 0, 117, 2, 113, 9, 106, 51, 99, 58 ], [ 0, 118, 4, 113, 11, 106, 53, 99, 60 ], [ 0, 119, 7, 113, 14, 106, 56, 99, 63 ], [ 0, 122, 6, 113, 13, 106, 55, 99, 62 ], [ 0, 123, 3, 113, 10, 106, 52, 99, 59 ], [ 0, 126, 5, 113, 12, 106, 54, 99, 61 ], [ 1, 125, 0, 113, 8, 106, 35, 92, 43 ] ] n = 8. Base cycles of a cyclic 9-cycle decomposition of the complete graph of order 145: w = 2. [ [ 0, 134, 2, 129, 10, 121, 58, 113, 66 ], [ 0, 135, 4, 129, 12, 121, 60, 113, 68 ], [ 0, 136, 6, 129, 14, 121, 62, 113, 70 ], [ 0, 137, 8, 129, 16, 121, 64, 113, 72 ], [ 0, 140, 7, 129, 15, 121, 63, 113, 71 ], [ 0, 141, 3, 129, 11, 121, 59, 113, 67 ], [ 0, 143, 1, 129, 9, 121, 57, 113, 65 ], [ 0, 144, 5, 129, 13, 121, 61, 113, 69 ] ] n = 2. Base cycles of a cyclic 10-cycle decomposition of the complete graph of order 41: w = 4. [ [ 0, 1, 39, 3, 37, 5, 35, 17, 33, 19 ], [ 0, 2, 39, 4, 37, 6, 35, 18, 33, 20 ] ] n = 4. Base cycles of a cyclic 10-cycle decomposition of the complete graph of order 81: w = 4. [ [ 0, 1, 77, 5, 73, 9, 69, 33, 65, 37 ], [ 0, 2, 77, 6, 73, 10, 69, 34, 65, 38 ], [ 0, 3, 77, 7, 73, 11, 69, 35, 65, 39 ], [ 0, 4, 77, 8, 73, 12, 69, 36, 65, 40 ] ] n = 6. Base cycles of a cyclic 10-cycle decomposition of the complete graph of order 121: w = 3. [ [ 0, 1, 115, 7, 109, 13, 103, 49, 97, 55 ], [ 0, 2, 115, 8, 109, 14, 103, 50, 97, 56 ], [ 0, 3, 115, 9, 109, 15, 103, 51, 97, 57 ], [ 0, 4, 115, 10, 109, 16, 103, 52, 97, 58 ], [ 0, 5, 115, 11, 109, 17, 103, 53, 97, 59 ], [ 0, 6, 115, 12, 109, 18, 103, 54, 97, 60 ] ] n = 8. Base cycles of a cyclic 10-cycle decomposition of the complete graph of order 161: w = 4. [ [ 0, 1, 153, 9, 145, 17, 137, 65, 129, 73 ], [ 0, 2, 153, 10, 145, 18, 137, 66, 129, 74 ], [ 0, 3, 153, 11, 145, 19, 137, 67, 129, 75 ], [ 0, 4, 153, 12, 145, 20, 137, 68, 129, 76 ], [ 0, 5, 153, 13, 145, 21, 137, 69, 129, 77 ], [ 0, 6, 153, 14, 145, 22, 137, 70, 129, 78 ], [ 0, 7, 153, 15, 145, 23, 137, 71, 129, 79 ], [ 0, 8, 153, 16, 145, 24, 137, 72, 129, 80 ] ]