Factorizations of complete graphs into brooms

Let rr and nn be positive integers with r<2nr<2n. A broom of order 2n2n is the union of the path on P2n−r−1P2n−r−1 and the star K1,rK1,r, plus one edge joining the center of the star to an endpoint of the path. It was shown by Kubesa (2005) [10] that the broom factorizes the complete graph K2nK2n for odd nn and r<⌊n2⌋. In this note we give a complete classification of brooms that factorize K2nK2n by giving a constructive proof for all r≤n+12 (with one exceptional case) and by showing that the brooms for r>n+12 do not factorize the complete graph K2nK2n.