7 research outputs found
Bipartitions of Highly Connected Tournaments
We show that if is a strongly -connected tournament,
there exists a partition of such that each of , and
is strongly -connected. This provides tournament analogues of two
partition conjectures of Thomassen regarding highly connected graphs.Comment: The main result of the current version strengthens the previous on
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
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