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A new proof of Delahan's induced-universality result

Jonathan Chappelon

Abstract

We give a short and self-contained proof of Delahan's theorem stating that every simple graph on $n$ vertices occurs as an induced subgraph of a Steinhaus graph on $\frac{n(n-1)}{2}+1$ vertices. This new proof is obtained by considering the notion of generating index sets for Steinhaus triangles.

A new proof of Delahan's induced-universality result

Abstract

We give a short and self-contained proof of Delahan's theorem stating that every simple graph on vertices occurs as an induced subgraph of a Steinhaus graph on vertices. This new proof is obtained by considering the notion of generating index sets for Steinhaus triangles.
Paper Structure (4 sections, 7 theorems, 51 equations, 5 figures)

This paper contains 4 sections, 7 theorems, 51 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Generating index sets
  3. Minors of binomial matrix
  4. Proof of Delahan's result

Key Result

Theorem 1.1

Let $n$ be a positive integer. We consider the $n$-subset of $\left\{1,2,\ldots,t_{n-1}+1\right\}$. Then, the linear map is an isomorphism. Therefore, any simple graph of order $n$ is isomorphic to an induced subgraph of a Steinhaus graph of order $t_{n-1}+1$.

Figures (5)

  • Figure 1: The Steinhaus triangle $\nabla (0010100)$ of size $7$
  • Figure 2: The Steinhaus graph $\mathrm{G}\!\left(0010100\right)$ and its adjacency matrix $\mathrm{M}\!\left(0010100\right)$
  • Figure 3: Generating index sets of $\mathcal{ST}\!\left(3\right)$
  • Figure 4: $3$-subsets of $\mathrm{T}_{3}$ that are not generating index sets of $\mathcal{ST}\!\left(3\right)$
  • Figure 5: Structure of the matrix $\bm{\mathrm{M}_{A_5}}$

Theorems & Definitions (14)

  • Theorem 1.1: Delahan Delahan:1998aa
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • ...and 4 more