Distinguishing Orthogonality Graphs

TL;DR

This paper investigates the properties of orthogonality graphs, establishing their determining number and bounds on their distinguishing number, which are key to understanding their symmetries and automorphisms.

Contribution

It determines the exact determining number of orthogonality graphs and provides bounds on their distinguishing number based on combinatorial parameters.

Findings

Det($ ext{Ω}_{2k}$) = 2^{2k-1}

Bounds on Dist($ ext{Ω}_{2k}$): 2 < Dist($ ext{Ω}_{2k}$) ≤ m when inom{m}{2} ≥ 2k

Provides insights into automorphisms and symmetry-breaking in orthogonality graphs

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 $\Omega_{2k}$ 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($\Omega_{2k}$) $= 2^{2k-1}$ and that if $\binom{m}{2} \geq 2k$ then $2<$ Dist($\Omega_{2k}$) $\leq m$.