Cycles in enhanced hypercubes

TL;DR

This paper studies cycle lengths in enhanced hypercubes, proving that each edge lies on cycles of all even lengths up to 2^n and certain odd lengths, revealing detailed cycle structure.

Contribution

It characterizes the cycle lengths that edges in enhanced hypercubes belong to, including all even lengths and specific odd lengths, extending understanding of their cycle properties.

Findings

Edges lie on all even cycles from 4 to 2^n

Edges lie on odd cycles from k+3 to 2^n-1 when k is even

Some edges are on shortest odd cycles of length k+1

Abstract

The enhanced hypercube $Q_{n,k}$ is a variant of the hypercube $Q_n$. We investigate all the lengths of cycles that an edge of the enhanced hypercube lies on. It is proved that every edge of $Q_{n,k}$ lies on a cycle of every even length from $4$ to $2^n$; if $k$ is even, every edge of $Q_{n,k}$ also lies on a cycle of every odd length from $k+3$ to $2^n-1$, and some special edges lie on a shortest odd cycle of length $k+1$.