On the Distinguishing Number of Cyclic Tournaments: Towards the Albertson-Collins Conjecture

TL;DR

This paper investigates the distinguishing number of cyclic tournaments, proving the Albertson-Collins Conjecture for various classes, including Paley tournaments, by analyzing substructure rigidity.

Contribution

It establishes conditions under which cyclic tournaments satisfy the Albertson-Collins Conjecture, extending known results and confirming the conjecture for all Paley tournaments.

Findings

Proves the conjecture for tournaments with rigid induced subtournaments.

Shows all Paley tournaments satisfy the conjecture.

Identifies conditions linking substructure rigidity to the conjecture.

Abstract

A distinguishing $r$-labeling of a digraph $G$ is a mapping $\lambda$ from the set of verticesof $G$ to the set of labels $\{1,\dots,r\}$ such that no nontrivial automorphism of $G$ preserves all the labels.The distinguishing number $D(G)$ of $G$ is then the smallest $r$ for which $G$ admits a distinguishing $r$-labeling.From a result of Gluck (David Gluck, Trivial set-stabilizers in finite permutation groups,{\em Can. J. Math.} 35(1) (1983), 59--67),it follows that $D(T)=2$ for every cyclic tournament~$T$ of (odd) order $2q+1\ge 3$.Let $V(T)=\{0,\dots,2q\}$ for every such tournament.Albertson and Collins conjectured in 1999that the canonical 2-labeling $\lambda^*$ given by$\lambda^*(i)=1$ if and only if $i\le q$ is distinguishing.We prove that whenever one of the subtournaments of $T$ induced by vertices $\{0,\dots,q\}$or $\{q+1,\dots,2q\}$ is rigid, $T$ satisfies Albertson-Collins Conjecture.Using this property, we prove that several classes of cyclic tournaments satisfy Albertson-Collins Conjecture.Moreover, we also prove that every Paley tournament satisfies Albertson-Collins Conjecture.