A Note on the Middle Levels Conjecture

TL;DR

This paper verifies the middle levels conjecture for k=18 by constructing a Hamiltonian cycle in the 37-dimensional hypercube using a new decomposition technique and an efficient algorithm, extending previous known cases.

Contribution

It introduces a novel decomposition method and an efficient algorithm to construct Hamiltonian cycles in the middle levels of hypercubes, confirming the conjecture for k=18.

Findings

Confirmed the conjecture for k=18 with a computer-assisted construction.

Developed a new decomposition technique for middle levels.

Created an efficient algorithm for ordering Narayana objects.

Abstract

The middle levels conjecture asserts that there is a Hamiltonian cycle in the middle two levels of $2k+1$-dimensional hypercube. The conjecture is known to be true for $k \leq 17$ [I.Shields, B.J.Shields and C.D.Savage, Disc. Math., 309, 5271--5277 (2009)]. In this note, we verify that the conjecture is also true for $k=18$ by constructing a Hamiltonian cycle in the middle two levels of 37-dimensional hypercube with the aid of the computer. We achieve this by introducing a new decomposition technique and an efficient algorithm for ordering the Narayana objects.