Bi-cyclic decompositions of complete graphs into spanning trees

We examine decompositions of complete graphs K4k+2K4k+2 into 2k+12k+1 isomorphic spanning trees. We develop a method of factorization based on a new type of vertex labelling, namely blended ρρ-labelling. We also show that for every k⩾1k⩾1 and every d,3⩽d⩽4k+1d,3⩽d⩽4k+1 there is a tree with diameter dd that decomposes K4k+2K4k+2 into 2k+12k+1 factors isomorphic to TT.