Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Zero Forcing Number of Twisted Hypercubes

Peter Collier, Jeannette Janssen

TL;DR

This work determines an explicit upper bound on the zero forcing number for a new family of twisted hypercubes, the minority cubes \(\hat{Q}_n\). By constructing forcing arc sets $\mathcal{F}_n$ that avoid chain twists, the authors certify zero forcing sets of size \(|S|=|V(\hat{Q}_n)|-|\mathcal{F}_n|=2^{n-1}-2^{n-3}+1$, for all \(n\ge3\). The construction proceeds inductively via two copies of smaller minority cubes connected by a bridge arc, yielding a recurrence \(|\mathcal{F}_n|=2|\mathcal{F}_{n-1}|+1\) and ensuring all chains have at most two arcs. They also establish a precise arc-count formula \(|\mathcal{F}_n|=2^{n-1}+2^{n-3}-1\) and describe the structural constraints that prevent chain twists, linking the arc-set properties to the zero forcing bound. The result highlights a nontrivial improvement over the standard hypercube bound and opens avenues for exploring lower bounds and other twisted-hypercube families.

Abstract

Twisted hypercubes are graphs that generalize the structure of the hypercube by relaxing the symmetry constraint while maintaining degree-regularity and connectivity. We study the zero forcing number of twisted hypercubes. Zero forcing is a graph infection process in which a particular colour change rule is iteratively applied to the graph and an initial set of vertices. We use the alternative framing of forcing arc sets to construct a family of twisted hypercubes of dimension k$\geq 3$ with zero forcing sets of size $2^{k-1}-2^{k-3}+1$, which is below the minimum zero forcing number of the hypercube.

The Zero Forcing Number of Twisted Hypercubes

TL;DR

This work determines an explicit upper bound on the zero forcing number for a new family of twisted hypercubes, the minority cubes . By constructing forcing arc sets that avoid chain twists, the authors certify zero forcing sets of size \(|S|=|V(\hat{Q}_n)|-|\mathcal{F}_n|=2^{n-1}-2^{n-3}+1$, for all . The construction proceeds inductively via two copies of smaller minority cubes connected by a bridge arc, yielding a recurrence and ensuring all chains have at most two arcs. They also establish a precise arc-count formula and describe the structural constraints that prevent chain twists, linking the arc-set properties to the zero forcing bound. The result highlights a nontrivial improvement over the standard hypercube bound and opens avenues for exploring lower bounds and other twisted-hypercube families.

Abstract

Twisted hypercubes are graphs that generalize the structure of the hypercube by relaxing the symmetry constraint while maintaining degree-regularity and connectivity. We study the zero forcing number of twisted hypercubes. Zero forcing is a graph infection process in which a particular colour change rule is iteratively applied to the graph and an initial set of vertices. We use the alternative framing of forcing arc sets to construct a family of twisted hypercubes of dimension k with zero forcing sets of size , which is below the minimum zero forcing number of the hypercube.
Paper Structure (7 sections, 14 theorems, 8 equations, 6 figures)

This paper contains 7 sections, 14 theorems, 8 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction
  4. Constructing $(\hat{Q}_4,\mathcal{F}_4)$
  5. Construction for $n>4$
  6. Proof of Main Result
  7. Future Work

Key Result

Theorem 1.3

chaintwist Let $G$ be a graph and $\mathcal{F}$ an arc set of $G$. Then $\mathcal{F}$ is a forcing arc set if and only if $\mathcal{F}$ does not contain a chain twist.

Figures (6)

  • Figure 1: An example of constructing $Q_3$ from two copies of $Q_2$ using the standard matching. Appended digits indicated in red.
  • Figure 2: An example of a 3-dimensional twisted hypercube constructed from two copies of $Q_2$ with twisted edges indicated in blue.
  • Figure 3: $(\hat{Q}_3,\mathcal{F}_3)$. The arc set $\mathcal{F}_3$ drawn on the graph $Q_3$.
  • Figure 4: The forcing arc set $\mathcal{F}_4$ indicated on the graph $\hat{Q}_4$ with twisted edges shown in red.
  • Figure 5: How the forcing arc set $\mathcal{F}_n$ is constructed from two copies of $(\hat{Q}_{n-1},\mathcal{F}_{n-1})$
  • ...and 1 more figures

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.5
  • ...and 17 more