We prove that every tournament with minimum out-degree at least 2k− 1 contains k disjoint 3-cycles. This provides additional support for the conjecture by Bermond and Thomassen that every digraph D of minimum out-degree 2k − 1 contains k vertex disjoint cycles. We also prove that for every > 0, when k is large enough, every tournament with minimum out-degree at least (1.5+ )k contains k disjoin...

Let A be a finite matrix with rational entries. We say that A is doubly image partition regular if whenever the set N of positive integers is finitely coloured, there exists ~x such that the entries of A~x are all the same colour (or monochromatic) and also, the entries of ~x are monochromatic. Which matrices are doubly image partition regular? More generally, we say that a pair of matrices (A,...

A tournament is a digraph in which, for each pair of distinct vertices v and w, either (v,w) or (w, v) is an edge, but not both. A tournament is regular if the in-degree is equal to the out-degree at each vertex. Let v1, v2, . . . , vn be the vertices of a labelled tournament and let d−j , d + j be the in-degree and out-degree of vj for 1 ≤ j ≤ n. d+j is also called the score of vj . Define δj ...

Let (x, y) be a specified arc in a k-regular bipartite tournament B. We prove that there exists a cycle C of length four through (x, y) in B such that B-C is hamiltonian.

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. In a recent article, the authors proved that a regular c-partite tournament with r ≥ 2 vertices in each partite set contains a cycle with exactly r− 1 vertices from each partite set, with exception of the case that c = 4 and r = 2. Here we wi...

We prove that every tree of order n 5 with three leaves is (n + 1)-unavoidable. More precisely, we prove that every tree A with three leaves of order n is contained in every tournament T of order n + 1 except if (T ; A) is (R 5 ; S + 3) or its dual, where R 5 is the regular tournament on ve vertices and S + 3 is the outstar of degree three, that is, the tree consisting of a root dominating thre...

We present some results on two-path convexity in clone-free regular multipartite tournaments. After proving a structural result for regular multipartite tournaments with convexly independent sets of a given size, we determine tight upper bounds for their size (called the rank) in clone-free regular bipartite and tripartite tournaments. We use this to determine tight upper bounds for the Helly a...

We prove that a regular tournament with n vertices has more than n 2 11.5 (1 − o(1)) pairwise arc-disjoint directed triangles. On the other hand, we construct regular tournaments with a feedback arc set of size less than n 2 8 , so these tournaments do not have n 8 pairwise arc-disjoint triangles. These improve upon the best known bounds for this problem.