Star-Forest Decompositions of Complete Graphs
<|reference_start|>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 $\frac{n}{2}+1$ 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.<|reference_end|>
@article{antić2024star-forest,
    title={Star-Forest Decompositions of Complete Graphs},
    author={Todor Anti'c, Jelena Gliv{s}i'c, Milan Milivojv{c}evi'c},
    journal={arXiv preprint arXiv:2402.11044},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.11044},
    primaryClass={math.CO cs.CG}
}