The vertex-pancyclicity of the simplified shuffle-cube and the vertex-bipancyclicity of the balanced shuffle-cube

TL;DR

This work analyzes cycle-embedding properties in two vertex-transitive shuffle-cube variants: the simplified shuffle-cube $SSQ_n$ and the balanced shuffle-cube $BSQ_n$. It establishes that $SSQ_n$ is vertex-pancyclic for $n\ge 6$ and $BSQ_n$ is vertex-bipancyclic for $n\ge 2$ by inductive constructions that combine small fixed cycles (lengths 16 and 32) with Hamiltonian paths in subcubes to realize any allowable cycle length through a given vertex or edge. The main technique relies on recursive decomposition into subcubes and careful assembly of cycles to cover all required lengths, ensuring vertex/edge coverage. These results reinforce the cycle-embedding capabilities of shuffle-cube variants, supporting their appeal for robust interconnection networks and suggesting avenues for fault-tolerant cycle embeddings.

Abstract

A graph $G$ $=$ $(V,E)$ is vertex-pancyclic if for every vertex $u$ and any integer $l$ ranging from $3$ to $|V|$, $G$ contains a cycle $C$ of length $l$ such that $u$ is on $C$. A bipartite graph $G$ $=$ $(V,E)$ is vertex-bipancyclic if for every vertex $u$ and any even integer $l$ ranging from $4$ to $|V|$, $G$ contains a cycle $C$ of length $l$ such that $u$ is on $C$. The simplified shuffle-cube and the balanced shuffle-cube, which are two variants of the shuffle-cube and are superior to shuffle-cube in terms of vertex-transitivity. In this paper, we show that the $n$-dimensional simplified shuffle-cube is vertex-pancyclic for $n\geqslant 6$, and the $n$-dimensional balanced shuffle-cube is vertex-bipancyclic for $n\geqslant 2$.

Figures (2)

• Figure 1: $SSQ_{6}$.
• Figure 2: $BSQ_{6}$.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 6
• Lemma 7
• Lemma 8