Decomposition of Cliques into $k$-Star-Forests

Abstract

A $k$-star-forest is a forest with at most $k$ connected components where each component is a star. Let $F_k(n)$ be the minimum integer such that the complete graph on $n$ vertices can be decomposed into $F_k(n)$ $k$-star-forests. Pach, Saghafian and Schnider showed that $F_2(n)=\lceil 3n/4 \rceil$. In this paper, we show that $F_3(n)=5n/9$ when $n$ is a multiple of 27. Further, for $k\ge 4$, we show that $F_k(n)=n/2+2$ when $n>2k$ and $n\equiv 4 \pmod{12}$. Our results disprove a conjecture of Pach, Saghafian and Schnider.