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Distinguishing Orthogonality Graphs

Debra Boutin, Sally Cockburn

Abstract

A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertices with $d$ labels so that only the trivial automorphism preserves the labels. The smallest such $d$ is the distinguishing number, Dist($G$). A subset of vertices $S$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The size of a smallest determining set for $G$ is called the determining number, Det($G$). The orthogonality graph $Ω_{2k}$ has vertices which are bitstrings of length $2k$ with an edge between two vertices if they differ in precisely $k$ bits. This paper shows that Det($Ω_{2k}$) $= 2^{2k-1}$ and that if $\binom{m}{2} \geq 2k$ then $2<$ Dist($Ω_{2k}$) $\leq m$.

Distinguishing Orthogonality Graphs

Abstract

A graph is said to be -distinguishable if there is a labeling of the vertices with labels so that only the trivial automorphism preserves the labels. The smallest such is the distinguishing number, Dist(). A subset of vertices is a determining set for if every automorphism of is uniquely determined by its action on . The size of a smallest determining set for is called the determining number, Det(). The orthogonality graph has vertices which are bitstrings of length with an edge between two vertices if they differ in precisely bits. This paper shows that Det() and that if then Dist() .

Paper Structure

This paper contains 11 sections, 18 theorems, 25 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Determining Sets and Distinguishing Labelings
  4. Orthogonality Graphs
  5. ${\rm Det}(\Omega_{2k})$ and a Lower Bound on ${\rm Dist}(\Omega_{2k})$
  6. Structure and $\widetilde{\Omega}_{2k}$
  7. Determining $\widetilde{\Omega}_{2k}$
  8. Distinguishing $\Omega_{2k}$
  9. Open Questions
  10. Acknowledgments
  11. Proof of Lemma \ref{['lem:Binom']}

Key Result

Lemma 1

Let $G$ be a graph, $\alpha \in {\rm Aut}(G)$, and $f:V(G) \to \{1, \dots, d\}$ a vertex labeling. Then $f$ is $d$-distinguishing if and only if $f \circ \alpha$ is $d$-distinguishing.

Figures (5)

  • Figure 1: $\widetilde{\Omega}_4$ with a 5-distinguishing labeling and $\Omega_4$ with a 4-distinguishing labeling
  • Figure 2: The weight of a neighbor of $\mathbf u$ is $m + k - 2t$.
  • Figure 3: Case 1: $|\text{supp}(\mathbf u) \cap \text{supp}(\mathbf w)| = 0$
  • Figure 4: Case 2: $r = |\text{supp}(\mathbf u) \cap \text{supp}(\mathbf w)|$ is odd.
  • Figure 5: Case 2: $r = |\text{supp}(\mathbf u) \cap \text{supp}(\mathbf w)|$ is even.

Theorems & Definitions (37)

  • Lemma 1
  • proof
  • Theorem \oldthetheorem
  • Corollary 1
  • proof
  • Definition 1
  • Lemma 2
  • Definition 2
  • Example 1
  • Example 2
  • ...and 27 more