A Perfect One-Factorisation of $K_{56}$

TL;DR

This paper proves that the smallest unresolved case of Kotzig's conjecture, which states that complete graphs can be decomposed into perfect matchings forming Hamilton cycles, is true for $K_{56}$.

Contribution

It confirms the existence of a perfect one-factorisation for $K_{56}$, resolving the smallest open case of Kotzig's conjecture.

Findings

Confirmed a perfect one-factorisation for $K_{56}$

Resolved the smallest unresolved case of Kotzig's conjecture

Supports the conjecture's validity for larger complete graphs

Abstract

In 1963, Anton Kotzig conjectured that for each $n \geq 2$ the complete graph $K_{2n}$ has a perfect one-factorisation (i.e., a decomposition into perfect matchings such that each pair of perfect matchings of the decomposition induces a Hamilton cycle). We affirmatively settle the smallest unresolved case for this conjecture.