Even tournaments and Hadamard tournaments

Local Tournaments and In - Tournaments

Preface Tournaments constitute perhaps the most well-studied class of directed graphs. One of the reasons for the interest in the theory of tournaments is the monograph Topics on Tournaments [58] by Moon published in 1968, covering all results on tournaments known up to this time. In particular, three results deserve special mention: in 1934 Rédei [60] proved that every tournament has a directe...

A Note on Even Cycles and Quasirandom Tournaments

A cycle C = {v1, v2, . . . , v1} in a tournament T is said to be even, if when walking along C, an even number of edges point in the wrong direction, that is, they are directed from vi+1 to vi. In this short paper, we show that for every fixed even integer k ≥ 4, if close to half of the k-cycles in a tournament T are even, then T must be quasi-random. This resolves an open question raised in 19...

t-Pancyclic Arcs in Tournaments

Let $T$ be a non-trivial tournament. An arc is emph{$t$-pancyclic} in $T$, if it is contained in a cycle of length $ell$ for every $tleq ell leq |V(T)|$. Let $p^t(T)$ denote the number of $t$-pancyclic arcs in $T$ and $h^t(T)$ the maximum number of $t$-pancyclic arcs contained in the same Hamiltonian cycle of $T$. Moon ({em J. Combin. Inform. System Sci.}, {bf 19} (1994), 207-214) showed that $...

Testing Tournaments

Ranking Tournaments

A tournament is an oriented complete graph. The feedback arc set problem for tournaments is the optimization problem of determining the minimum possible number of edges of a given input tournament T whose reversal makes T acyclic. Ailon, Charikar and Newman showed that this problem is NP-hard under randomized reductions. Here we show that it is in fact NP-hard. This settles a conjecture of Bang...