Construction of 2-factors in the middle layer of the discrete cube

TL;DR

This paper presents an inductive method to construct extensive 2-factors in the middle layer graph, analyzing how parameter choices influence cycle structures, advancing understanding of middle layer graph properties.

Contribution

It introduces a novel inductive construction of 2-factors in the middle layer graph applicable for all n ≥ 1, exploring parameter effects on cycle configurations.

Findings

Constructed large families of 2-factors in the middle layer graph.

Analyzed how parameter choices affect cycle numbers and lengths.

Provided insights into the structure of middle layer graphs.

Abstract

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit. In this work we present an inductive construction of a large family of 2-factors in the middle layer graph for all $n\geq 1$. We also investigate how the choice of certain parameters used in the construction affects the number and lengths of the cycles in the resulting 2-factor.