Twin-star hypothesis and cycle-free $d$-partitions of $K_{2d}$ ]{Twin-star hypothesis and cycle-free $d$-partitions of $K_{2d}$

TL;DR

This paper explores a conjecture linking cycle-free $d$-partitions of complete graphs to the twin-star hypothesis, providing evidence for the conjecture in specific cases and discussing related algebraic maps.

Contribution

It establishes the equivalence between a conjecture on cycle-free partitions and the twin-star hypothesis, and verifies the conjecture for the case $d=4$.

Findings

Conjecture holds for $d=4$ case.

Equivalence between the conjecture and twin-star hypothesis.

Discussion of the determinant-like map $det^{S^2}$.

Abstract

In this paper we study an equivalence relation defined on the set of cycle-free $d$-partitions of the complete graph $K_{2d}$. We discuss a conjecture which states that this equivalence relation has only one equivalence class, and show that the conjecture is equivalent with the so called twin-star hypothesis. We check the conjecture in the case $d=4$ and disuses how this relates to the determinant-like map $det^{S^2}$.