Factorizations of complete graphs into brooms

Let r and n be positive integers with r < 2n. A broom of order 2n is the union of the path on P2n-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-2n for odd n and r < [n/2]. In this note we give a complete classification of brooms that factorize K2n 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-2n. (C) 2011 Elsevier B.V. All rights reserved.