A distinguishing $r$-labeling of a digraph $G$ is a mapping $lambda$ fromthe set of verticesof $G$ to the set of labels ${1,dots,r}$ such that nonontrivial automorphism of $G$ preserves all the labels.The distinguishingnumber $D(G)$ of $G$ is then the smallest $r$ for which $G$ admits adistinguishing $r$-labeling.From a result of Gluck (David Gluck, Trivialset-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$2p+1ge 3$.Let $V(T)={0,dots,2p}$ for every such tournament.Albertson andCollins conjectured in 1999that the canonical 2-labeling $lambda^*$ givenby$lambda^*(i)=1$ if and only if $ile p$ is distinguishing.We prove thatwhenever one of the subtournaments of $T$ induced by vertices ${0,dots,p}$or${p+1,dots,2p}$ is rigid, $T$ satisfies Albertson-Collins Conjecture.Usingthis property, we prove that several classes of cyclic tournaments satisfyAlbertson-Collins Conjecture.Moreover, we also prove that every Paleytournament satisfies Albertson-Collins Conjecture.
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机译：的verticesof $ G $ A $区分R $一个有向图$ G $的 - 标号被映射$ 拉姆达$ fromthe设置到标签集$ {1，点，R } $使得$的nonontrivial构G $保留所有labels.The distinguishingnumber $ d（G）$ $ G $的是然后最小$ R $为哪些$ G $承认adistinguishing $ R $ -labeling.From格鲁克（大卫格鲁克，Trivialset-的结果在有限置换群稳定剂，{ EM可以。J.数学} 35（1）（1983），59--67），它遵循$ d（T）= 2 $每环状比赛〜$ T $ （奇数）顺序$ 2P + 1 GE 3 $。让$ V（T）= {0，点，2P } $用于推测1999that规范2-标记$ 拉姆达每一个这样的tournament.Albertson andCollins ^ * $ givenby $ 拉姆达^ *（1）= 1 $当且仅当$ I 文件p $是distinguishing.We证明$ T $的subtournaments由顶点$诱导 {0，点thatwhenever一个，对} $或者$ {p + 1，点，2P } $是刚性的，$ T $满足艾伯森-柯林斯Conjecture.Usingthis属性，我们证明了几类环状比赛satisfyAlbertson-柯林斯Conject的ure.Moreover，我们也证明了每Paleytournament满足艾伯森柯林斯猜想。
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