Factorizations of complete graphs into brooms

Let r and n be positive integers with r < 2 n . A broom of order 2 n is the union of the path on P 2 n − r − 1 and the star K 1 , 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 K 2 n for odd n and r < ⌊ n 2 ⌋ . In this note we give a complete classification of brooms that factorize K 2 n by giving a constructive proof for all r ≤ n + 1 2 (with one exceptional case) and by showing that the brooms for r > n + 1 2 do not factorize the complete graph K 2 n .