Factorizations of complete graphs into brooms

Abstract

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 K1,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 K2n for odd n and View the MathML source. In this note we give a complete classification of brooms that factorize K2n by giving a constructive proof for all View the MathML source (with one exceptional case) and by showing that the brooms for View the MathML source do not factorize the complete graph K2n.