Distinguishing threshold of graphs

TL;DR

This paper characterizes the distinguishing threshold of graphs, explores its properties for various graph classes, and computes it for generalized Johnson graphs, revealing relationships with automorphisms and symmetry-breaking.

Contribution

It provides a complete characterization of the distinguishing threshold for disconnected graphs and computes it for generalized Johnson graphs, linking it to automorphism groups.

Findings

or very positive integer kxcept 2, infinitely many graphs have istinguishing threshold k.

istinguishing threshold quals 2 only for graphs with 2 vertices.

or Johnson graphs, the threshold is omputed explicitly, revealing structural symmetry properties.

Abstract

A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such a coloring, and the distinguishing threshold of $G$, denoted by $\theta(G)$, is the minimum number $k$ such that every $k$-coloring of $G$ is distinguishing. As an alternative definition, $\theta(G)$ is one more than the maximum number of cycles in the cycle decomposition of automorphisms of $G$. In this paper, we characterize $\theta (G)$ when $G$ is disconnected. Afterwards, we prove that, although for every positive integer $k\neq 2$ there are infinitely many graphs whose distinguishing thresholds are equal to $k$, we have $\theta(G)=2$ if and only if $\vert V(G)\vert =2$. Moreover, we show that if $\theta(G)=3$, then either $G$ is isomorphic to one of the four graphs on~3 vertices or it is of order $2p$, where $p\neq 3,5$ is a prime number. Furthermore, we prove that $\theta(G)=D(G)$ if and only if $G$ is asymmetric, $K_n$ or $\overline{K_n}$. Finally, we consider all generalized Johnson graphs, $J(n,k,i)$, which are the graphs on all $k$-subsets of $\{1,\ldots , n\}$ where two vertices $A$ and $B$ are adjacent if $|A\cap B|=k-i$. After studying their automorphism groups and distinguishing numbers, we calculate their distinguishing thresholds as $\theta(J(n,k,i))={n\choose k} - {n-2\choose k-1}+1$, unless $ k=\frac{n}{2}$ and $i\in\{ \frac{k}{2} , k\}$ in which case we have $\theta(J(n,k,i))={n\choose k}$.