Cycles and Paths Embedded in Varietal Hypercubes

TL;DR

This paper investigates the cycle and path structures in varietal hypercubes, revealing that most edges and vertex pairs are part of cycles and paths of various lengths, highlighting their rich connectivity properties.

Contribution

It establishes comprehensive cycle and path length properties in varietal hypercubes, extending understanding beyond traditional hypercube structures.

Findings

Edges are in cycles of all lengths from 4 to 2^n except 5.

Vertex pairs at distance d connect via paths of all lengths from d to 2^n-1, with specific exceptions.

Varietal hypercubes exhibit extensive cycle and path diversity, enhancing their network robustness.

Abstract

The varietal hypercube $VQ_n$ is a variant of the hypercube $Q_n$ and has better properties than $Q_n$ with the same number of edges and vertices. This paper shows that every edge of $VQ_n$ is contained in cycles of every length from 4 to $2^n$ except 5, and every pair of vertices with distance $d$ is connected by paths of every length from $d$ to $2^n-1$ except 2 and 4 if $d=1$.