A distinguishing r-labeling of a digraph G is a mapping λ from
the set of verticesof G to the set of labels {1,…,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≥3.Let V(T)={0,…,2q} for every such tournament.Albertson and
Collins conjectured in 1999that the canonical 2-labeling λ∗ given
byλ∗(i)=1 if and only if i≤q is distinguishing.We prove that
whenever one of the subtournaments of T induced by vertices {0,…,q}or
{q+1,…,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