Related papers: Oriented Trees in Digraphs without Oriented $4$-cy…

It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an…

The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum…

We prove that for every set $S$ of vertices of a directed graph $D$, the maximum number of vertices in $S$ contained in a collection of vertex-disjoint cycles in $D$ is at least the minimum size of a set of vertices that hits all cycles…

Nikiforov (LAA, 2010) conjectured that for given integer $k$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu(G)\geq \mu(S_{n,k})$ contains all trees of order $2k+2$, unless $G=S_{n,k}$, where $S_{n,k}=K_k\vee…

Let $D$ be a strong digraph on $n=2m+1\geq 5$ vertices. In this paper we show that if $D$ contains a cycle of length $n-1$, then $D$ has also a cycle which contains all vertices with in-degree and out-degree at least $m$ (unless some…

Mader [J. Graph Theory 65 (2010) 61-69] conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree…

A class of acyclic digraphs $\mathscr{C}$ is linearly unavoidable if there exists a constant $c$ such that every digraph $D\in \mathscr{C}$ is contained in all tournaments of order $c\cdot |V(D)|$. The class of all acyclic digraphs is not…

The {\sc 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 improve known parameterized algorithms and…

Let $D$ be a digraph. A stable set $S$ of $D$ and a path partition $\mathcal{P}$ of $D$ are orthogonal if every path $P \in \mathcal{P}$ contains exactly one vertex of $S$. In 1982, Berge defined the class of $\alpha$-diperfect digraphs. A…

We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that…

A classical result by Otter shows that the complete graph has an exponential number of non-isomorphic spanning trees. This was recently extended by Lee to every almost regular graph of sufficiently large degree. In this paper, we consider…

In 1985, Mader conjectured that for every acyclic digraph $F$ there exists $K=K(F)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of $F$. This conjecture remains widely open, even for digraphs $F$…

Let $D$ be an directed graph on $p\geq 10$ vertices with minimum degree at least $p-1$ and minimum semi-degree at least $ p/2 -1$. We present a detailed proof of the following result [13]: The digraph $D$ is pancyclic, unless some extremal…

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is…

We show that every directed graph with minimum out-degree at least $18k$ contains at least $k$ vertex disjoint cycles. This is an improvement over the result of Alon who showed this result for digraphs of minimum out-degree at least $64k$.…

An oriented graph $D$ is converse invariant if, for any tournament $T$, the number of copies of $D$ in $T$ is equal to that of its converse $-D$. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph…

We conjecture that every $n$-vertex graph of minimum degree at least $\frac k2$ and maximum degree at least $2k$ contains all trees with $k$ edges as subgraphs. We prove an approximate version of this conjecture for trees of bounded degree…

An oriented graph is a directed graph which can be obtained from a simple undirected graph by orienting its edges. In this paper we show that any oriented graph G on n vertices with minimum indegree and outdegree at least (1/2-o(1))n…

A tree-coloring of a maximal planar graph is a proper vertex $4$-coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph $G$ is tree-colorable if $G$ has a tree-coloring. In this article,…

Let $ H $ be a multi-digraph on $ h $ vertices with $ q $ arcs. An \textbf{$H$-subdivision} in a digraph $D$ is a subdigraph obtained by replacing every arc $uv$ of $H$ with a path from $u$ to $v$ in $D$ such that these paths are pairwise…