Directed Cycles with Two Chords and Strong Spanning Directed Subgraphs with Few Arcs

Spanning k-arc-strong subdigraphs with few arcs in k-arc-strong tournaments

Given a k-arc-strong tournament T , we estimate the minimum number of arcs possible in a k-arc-strong spanning subdigraph of T . We give a construction which shows that for each k ≥ 2 there are tournaments T on n vertices such that every k-arc-strong spanning subdigraph of T contains at least nk+ k(k−1) 2 arcs. In fact, the tournaments in our construction have the property that every spanning s...

Sparse highly connected spanning subgraphs in dense directed graphs

Mader (1985) proved that every strongly $k$-connected $n$-vertex digraph contains a strongly $k$-connected spanning subgraph with at most $2kn - 2k^2$ edges, and this is sharp. For dense strongly $k$-connected digraphs, the bound can be significantly improved for dense strongly $k$-connected digraphs. Let $\overline{\Delta}(D)$ be the maximum degree of the complement of the underlying undirecte...

Long cycles in subgraphs of (pseudo)random directed graphs

We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 0 < γ < 1/2 we find a constant c = c(γ) such that the following holds. Let G = (V,E) be a (pseudo)random directed graph on n vertices and with at least a linear number of edges, and let G′ be a subgraph of G with (1/2 + γ)|E| edges. Then G′ contains a directe...

Two-connected Spanning Subgraphs with at Most

Spanning directed trees with many leaves

The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that – every strongly connected n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) −...