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Cycles in Oriented 3-graphs
An oriented 3-graph consists of a family of triples (3-sets), each of which
is given one of its two possible cyclic orientations. A cycle in an oriented
3-graph is a positive sum of some of the triples that gives weight zero to each
2-set.
Our aim in this paper is to consider the following question: how large can
the girth of an oriented 3-graph (on vertices) be? We show that there exist
oriented 3-graphs whose shortest cycle has length : this
is asymptotically best possible. We also show that there exist 3-tournaments
whose shortest cycle has length , in complete contrast
to the case of 2-tournaments.Comment: 12 page
Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths
In 1982 Thomassen asked whether there exists an integer f(k,t) such that
every strongly f(k,t)-connected tournament T admits a partition of its vertex
set into t vertex classes V_1,...,V_t such that for all i the subtournament
T[V_i] induced on T by V_i is strongly k-connected. Our main result implies an
affirmative answer to this question. In particular we show that f(k,t) = O(k^7
t^4) suffices. As another application of our main result we give an affirmative
answer to a question of Song as to whether, for any integer t, there exists an
integer h(t) such that every strongly h(t)-connected tournament has a 1-factor
consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t)
= O(t^5) suffices.Comment: final version, to appear in Combinatoric
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