Star-Forest Decompositions of Complete Graphs

We deal with the problem of decomposing a complete geometric graph into plane star-forests. In particular, we disprove a recent conjecture by Pach, Saghafian and Schnider by constructing for each n a complete geometric graph on n vertices which can be decomposed into n2+1 plane star-forests. Additionally we prove that for even n, every decomposition of complete abstract graph on n vertices into n2+1 star-forests is composed of a perfect matching and n2 star-forests with two edge-balanced components, which we call broken double stars.