On the Distinguishing Number of Cyclic Tournaments: Towards the Albertson-Collins Conjecture
Résumé
A distinguishing $r$-labeling of a digraph $G$ is a mapping $\lambda$ from the set of vertices
of $G$ 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 1999
that 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.
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