Splitting a tournament into two subtournaments with given minimum outdegree

A {\it $(k_1,k_2)$-outdegree-splitting} of a digraph $D$ is a partition $(V_1,V_2)$ of its vertex set such that $D[V_1]$ and $D[V_2]$ have minimum outdegree at least $k_1$ and $k_2$, respectively. We show that there exists a minimum function $f_T$ such that every tournament of minimum outdegree at least $f_T(k_1,k_2)$ has a $(k_1,k_2)$-outdegree-splitting, and $f_T(k_1,k_2) \leq k_1^2/2+3k_1/2 +k_2+1$. We also show a polynomial-time algorithm that finds a $(k_1,k_2)$-outdegree-splitting of a tournament if one exists, and returns 'no' otherwise. We give better bound on $f_T$ and faster algorithms when $k_1=1$.Un {\it $(k_1,k_2)$-partage} d'un digraphe $D$ est une partition $(V_1,V_2)$ de son ensemble de sommets telle que $D[V_1]$ et $D[V_2]$ soient de degréß sortant minimum au moins $k_1$ et $k_2$, respectivement. Nous établissons l'existence d'une fonction (minimum) $f_T$ telle que tout tournoi de degré sortant minimum au moins $f_T(k_1,k_2)$ a un $(k_1,k_2)$-partage, et que $f_T(k_1,k_2) \leq k_1^2/2+3k_1/2 +k_2+1$. Nous donnons également un algorithme en temps polynomial qui trouve un $(k_1,k_2)$-partage d'un tournoi s'il en existe un et renvoie 'non' sinon. Nous donnons de meilleures bornes sur $f_T$ et des algorithmes plus rapides pour $k_1=1$

Complementary Cycles Containing Prescribed Vertices in Tournaments

We prove that if T is a tournament on n vertices and x; y are distinct vertices of T with the property that T remains 2-connected if we delete the arc between x and y, then there exist disjoint 3-cycles C x ; C y such that x 2 V (C x ) and y 2 V (C y ). This is best possible in terms of the connectivity assumption. Using this we prove that under the same assumptions T also contains complementary cycles C 0 x ; C 0 y (i.e. V (C 0 x )[V (C 0 y ) = V (T ) and V (C 0 x ) " V (C 0 y ) = ;) such that x 2 V (C 0 x ) and y 2 V (C 0 y ) for every choice of distinct vertices x; y 2 V (T ). Again this is best possible in terms of the connectivity assumption. It is a trivial consequence of our result that one can decide in polynomial time whether a given tournament T with special vertices x; y contains disjoint cycles C x ; C y as above. This problem is NP-complete for general digraphs and furthermore there is no degree of strong connectivity which suffices to guarantee such cyc..