Mader proved that every strongly k-connected n-vertex digraph contains a strongly k-connected spanning subgraph with at most 2kn - 2k2 edges, where equality holds for the complete bipartite digraph DKk,n-k. For dense strongly k-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly k-connected n-vertex digraph D contains a strongly k-connected spanning subgraph with at most kn + 800k(k + Δ(D)) edges, where Δ(D) denotes the maximum degree of the complement of the underlying undirected graph of a digraph D. Here, the additional term 800k(k + Δ(D)) is tight up to multiplicative and additive constants. As a corollary, this implies that every strongly k-connected n-vertex semicomplete digraph contains a strongly k-connected spanning subgraph with at most kn + 800k2 edges, which is essentially optimal since 800k2 cannot be reduced to the number less than k(k - 1)/2.

We also prove an analogous result for strongly k-arc-connected directed multigraphs. Both proofs yield polynomial-time algorithms.

For an orientation H with n vertices, let T(H) denote the maximum possible number of labelled copies of H in an n-vertex tournament. It is easily seen that T(H) ≥ n!/2e(H), as the latter is the expected number of such copies in a random tournament. For n odd, let R(H) denote the maximum possible number of labelled copies of H in an n-vertex regular tournament. In fact, Adler, Alon and Ross proved that for H=Cn, the directed Hamilton cycle, T(Cn) ≥ (e−o(1))n!/2n, and it was observed by Alon that already R(Cn) ≥ (e−o(1))n!/2n. Similar results hold for the directed Hamilton path Pn. In other words, for the Hamilton path and cycle, the lower bound derived from the expectation argument can be improved by a constant factor. In this paper we significantly extend these results, and prove that they hold for a larger family of orientations H which includes all bounded-degree Eulerian orientations and all bounded-degree balanced orientations, as well as many others. One corollary of our method is that for any fixed k, every k-regular orientation H with n vertices satisfies T(H) ≥ (ek−o(1))n!/2e(H), and in fact, for n odd, R(H) ≥ (ek−o(1))n!/2e(H).

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph Kn and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament Tk on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of Tk; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1))log2n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 − o(1))log2n Breaker can prevent the underlying graph of Maker's digraph from containing a k-clique. Moreover, the precise value of our lower bound differs from the upper bound only by an additive constant of 12.

We also discuss the question of whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two ‘clever’ players and the game played by two ‘random’ players, is supported by the tournament game. It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid.

Finally, we consider the orientation game version of the tournament game, where Maker wins the game if the final digraph – also containing the edges directed by Breaker – possesses a copy of Tk. We prove that in that game Breaker has a winning strategy for k = (4 + o(1))log2n.

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k-graphs: projective limits and crossed products by finitely generated free abelian groups.

Hajnal and Szemerédi proved that every graph G with |G| = ks and δ(G)⩾ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph $\vv G$ with |$\vv G$| = ks and δ($\vv G$) ⩾ 2k(s − 1) − 1 contains k disjoint transitive tournaments on s vertices, where δ($\vv G$)= minv∈V($\vv G$)d−(v)+d+(v). Our result implies the Hajnal–Szemerédi theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.

A minimum feedback arc set of a directed graph G is a smallest set of arcs whose removal makes G acyclic. Its cardinality is denoted by β(G). We show that a simple Eulerian digraph with n vertices and m arcs has β(G) ≥ m2/2n2+m/2n, and this bound is optimal for infinitely many m, n. Using this result we prove that a simple Eulerian digraph contains a cycle of length at most 6n2/m, and has an Eulerian subgraph with minimum degree at least m2/24n3. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in Eulerian digraphs.

Given a (multi)digraph H, a digraph D is H-linked if every injective function ι:V(H) → V(D) can be extended to an H-subdivision. In this paper, we give sharp degree conditions that ensure a sufficiently large digraph D is H-linked for arbitrary H. The notion of an H-linked digraph extends the classes of m-linked, m-ordered and strongly m-connected digraphs.

First, we give sharp minimum semi-degree conditions for H-linkedness, extending results of Kühn and Osthus on m-linked and m-ordered digraphs. It is known that the minimum degree threshold for an undirected graph to be H-linked depends on a partition of the (undirected) graph H into three parts. Here, we show that the corresponding semi-degree threshold for H-linked digraphs depends on a partition of H into as many as nine parts.

We also determine sharp Ore–Woodall-type degree-sum conditions ensuring that a digraph D is H-linked for general H. As a corollary, we obtain (previously undetermined) sharp degree-sum conditions for m-linked and m-ordered digraphs.

Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if, at time t, instead of a random direction one is allowed to choose the orientation of et, then whether or not it is possible to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p. the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.