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

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

Edge disjoint Hamiltonian cycles in highly connected tournaments

Thomassen conjectured that there is a function f(k) such that every strongly f(k)-connected tournament contains k edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kühn, Lapinskas, Osthus, and Patel who showed that f(k) ≤ O(k 2 (logk) 2 ) and conjectured that there is a constant C such that f(k) ≤ Ck 2 . We prove this conjecture. As a second application of our methods we answer a question of Thomassen about spanning linkages in highly connected tournaments

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 $k_1=1$

Extremal problems on graphs, directed graphs and hypergraphs

This thesis is concerned with extremal problems on graphs and similar structures. We first study degree conditions in uniform hypergraphs that force matchings of various sizes. Our main result in this area improves bounds of Markstrom and Rucinski on the minimum d-degree which forces a perfect matching in a k-uniform hypergraph on n vertices. We then study connectivity conditions in tournaments that ensure the existence of partitions of the vertex set that satisfy various properties. In 1982 Thomassen asked whether every sufficiently strongly connected tournament T admits a partition of its vertex set into t vertex classes such that the subtournament induced on T by each class is strongly k-connected. Our main result in this area implies an affirmative answer to this question. Finally we investigate the typical structure of graphs and directed graphs with some forbidden subgraphs. We answer a question of Cherlin by finding the typical structure of triangle-free oriented graphs. Moreover, our results generalise to forbidden transitive tournaments and forbidden oriented cycles of any order, and also apply to digraphs. We also determine, for all k>5, the typical structure of graphs that do not contain an induced 2k-cycle. This verifies a conjecture of Balogh and Butterfield