Edge bicoloring problems and odd graphs

An approach to edge bicoloring problems for biregular bipartite graphs is applied to graphs arising from cyclic groups as well as from odd graphs. In the case of the Petersen graph, this incidentally allows solutions to the Great Circle Challenge puzzle. On the other hand, for odd n = 2k+1 > 1, a one-to-one correspondence from the family of n-cycles of Kn onto the family Ok of n-cycles of the odd graph Ok+1 becomes apparent. This takes, for odd l with 3 ≤ l ≤ ⌊n/2⌋, to a graph having Ok as vertex set and an edge between any two vertices if the symmetric difference of the n-cycles they represent is an l-cycle. A total of k− 1 such graphs having pairwise orthogonal 1-factorizations have their union as a 1-factorable graph with n(k − 1) 1-factors.