Star-Forest Decompositions of Complete Graphs

TL;DR

This paper studies decomposing complete geometric graphs into plane star-forests, disproves a recent conjecture by constructing specific decompositions, and characterizes the structure of such decompositions for even n.

Contribution

It disproves a conjecture by providing explicit constructions and characterizes the structure of decompositions into star-forests for even n.

Findings

Constructed complete geometric graphs decomposable into  + 1 plane star-forests.

Disproved the conjecture by Pach, Saghafian, and Schnider.

Characterized decompositions of complete abstract graphs into star-forests for even n.

Abstract

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 $\frac{n}{2}+1$ plane star-forests. Additionally we prove that for even $n$, every decomposition of complete abstract graph on $n$ vertices into $\frac{n}{2}+1$ star-forests is composed of a perfect matching and $\frac{n}{2}$ star-forests with two edge-balanced components, which we call broken double stars.