Distinguishing Tournaments with Small Label Classes

TL;DR

This paper investigates the minimum label class sizes needed for distinguishing labelings in tournaments, constructing specific examples and establishing bounds that answer open questions in graph automorphism theory.

Contribution

It constructs a family of tournaments with large minimum label class sizes and establishes bounds on label class sizes for general tournaments, addressing open problems.

Findings

Constructed tournaments with minimum label class size at least half the order.

Proved upper bounds on label class sizes for all tournaments.

Established lower bounds for specific tournament families.

Abstract

A $d$-distinguishing vertex (arc) labeling of a digraph is a vertex (arc) labeling using $d$ labels that is not preserved by any nontrivial automorphism. Let $\rho(T)$ ($\rho'(T)$) be the minimum size of a label class in a 2-distinguishing vertex (arc) labeling of a tournament $T$. Gluck's Theorem implies that $\rho(T) \le \lfloor n/2 \rfloor$ for any tournament $T$ of order $n$. In this paper we construct a family of tournaments $\cal H$ such that $\rho(T) \ge \lfloor n/2 \rfloor$ for any order $n$ tournament in $\cal H$. Additionally, we prove that $\rho'(T) \le \lfloor 7n/36 \rfloor + 3$ for any tournament $T$ of order $n$ and $\rho'(T) \ge \lceil n/6 \rceil$ when $T \in {\cal H}$ and has order $n$. These results answer some open questions stated by Boutin.