Great Circle Challenge and Odd Graphs

Edge (5, 3)-colorings of a (5, 3)-regular bipartite graph relating 5cycles and vertices of the Petersen graph lead to solutions of the Great Circle Challenge. This takes, for odd n = 2k + 1 > 1, to a canonical 1-1 correspondence from the family of n-cycles of the complete graph Kn onto the family Ck of n-cycles of the odd graph Ok+1 and onto the family of their pullback 2n-cycles in the middle-levels graph on n symbols. It also takes, for odd l such that 3 ≤ l ≤ ⌊n/2⌋, to a graph Γk having Ck as vertex set and an edge between any two vertices if the symmetric difference of the n-cycles they represent is an l-cycle. This yields a total of k− 1 graphs Γk on Ck possessing pairwise orthogonal 1-factorizations, so their union is a 1-factorable graph with n(k − 1) 1-factors.