Embedding tournaments in simple 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...

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 $...

Simple tournaments and sharply transitive groups

If the relation E is transitive, i.e., if (x, y) E E and (y, z) E E imply that (x, z) E E, we speak of a transitive tournament or a total order. Clearly (Z, <) is a total order. We tend to visualize tournaments by considering every edge (x, y) E E as an arrow leading from x to y. In this sense every tournament can also be considered as a complete graph in which every edge is oriented in some di...

Simple One - Point Extensions of Tournaments

x->y,x=y,y->x holds. Here x -+ y expresses the fact that {x, y} e -+ and we sometimes write this in the alternative form y <--x . Extending the notation to subsets of T we write A --+ B or B is a subtournament of .l, and i is an extension of ,T', if T' e T and -+' is the restriction of --+ to T' ; we will usually writ...

Testing Tournaments