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

Kahina Meslem (1) , Eric Sopena (2)

1 Laboratoire de Recherche Opérationnelle et Mathématiques de la Décision - LAROMAD (Alger, Algérie)
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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. Albertson and Collins conjectured in 1999 that $D(T)\le 2$ for every cyclic tournament~$T$ of (odd) order $2p+1\ge 3$, with $V(T)=\{0,\dots,2p\}$, and, more precisely, that the canonical 2-labeling $\lambda^*$ given by $\lambda^*(i)=1$ if and only if $i\le p$ is distinguishing. We prove that whenever one of the subtournaments of $T$ induced by vertices $\{0,\dots,p\}$ or $\{p+1,\dots,2p\}$ 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.

Mots clés: Distinguishing number; Automorphism group; Cyclic tournament; Albertson-Collins Conjecture.

Domaines: Mathématique discrète [cs.DM]

Kahina Meslem, Eric Sopena. On the Distinguishing Number of Cyclic Tournaments: Towards Albertson-Collins Conjecture. 2016. ⟨hal-01353772v1⟩