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

A distinguishing r-labeling of a digraph G is a mapping λ from the set of vertices of 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, Can. J. Math. 35(1) (1983), 59–67), it follows that D(T ) = 2 for every cyclic tournament T of (odd) order 2p+1 ≥ 3. Let V (T ) = {0, . . . , 2p} for every such tournament. Albertson and Collins conjectured in 1999 that the canonical 2-labeling λ given by λ(i) = 1 if and only if i ≤ p is distinguishing. We prove that whenever one of the subtournaments of T induced by vertices {0, . . . , p} or {p+ 1, . . . , 2p} 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.