136 research outputs found
Compound Node-Kayles on Paths
In his celebrated book "On Number and Games" (Academic Press, New-York,
1976), J.H. Conway introduced twelve versions of compound games. We analyze
these twelve versions for the Node-Kayles game on paths. For usual disjunctive
compound, Node-Kayles has been solved for a long time under normal play, while
it is still unsolved under mis\`ere play. We thus focus on the ten remaining
versions, leaving only one of them unsolved.Comment: Theoretical Computer Science (2009) to appea
On the Distinguishing Number of Cyclic Tournaments: Towards the Albertson-Collins Conjecture
A distinguishing -labeling of a digraph is a mapping from
the set of verticesof to the set of labels such that no
nontrivial automorphism of preserves all the labels.The distinguishing
number of is then the smallest for which admits a
distinguishing -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 for every cyclic tournament~ of (odd) order
.Let for every such tournament.Albertson and
Collins conjectured in 1999that the canonical 2-labeling given
by if and only if is distinguishing.We prove that
whenever one of the subtournaments of induced by vertices or
is rigid, 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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