Related papers: Packing spanning arborescences with extra large on…

An \emph{out-tree (in-tree)} is an oriented tree where every vertex except one, called the \emph{root}, has in-degree (out-degree) one. An \emph{out-branching $B^+_u$ (in-branching $B^-_u$)} of a digraph $D$ is a spanning out-tree (in-tree)…

We show that for any $\varepsilon>0$ and $\Delta\in\mathbb{N}$, there exists $\alpha>0$ such that for sufficiently large $n$, every $n$-vertex graph $G$ satisfying that $\delta(G)\geq\varepsilon n$ and $e(X, Y)>0$ for every pair of disjoint…

Let $G=(V,E)$ and $H$ be two graphs. Packing problem is to find in $G$ the largest number of independent subgraphs each of which is isomorphic to $H$. Let $U\subset{V}$. If the graph $G-U$ has no subgraph isomorphic to $H$, $U$ is a cover…

The $d$-distance $p$-packing domination number $\gamma_d^p(G)$ of $G$ is the minimum size of a set of vertices of $G$ which is both a $d$-distance dominating set and a $p$-packing. In 1994, Beineke and Henning conjectured that if $d\ge 1$…

Let $ D $ be a finite digraph, and let $ V_0,\dots,V_{k-1} $ be nonempty subsets of $ V(D) $. The (strong form of) Edmonds' branching theorem states thatthere are pairwise edge-disjoint spanning branchings $ \mathcal{B}_0,\dots,…

Assume that the edges of the complete graph $K_n$ are given independent uniform $[0,1]$ edges weights. We consider the expected minimum total weight $\mu_k$ of $k\geq 2$ edge disjoint spanning trees. When $k$ is large we show that…

Given a graph $G$ and a digraph $D$ whose vertices are the edges of $G$, we investigate the problem of finding a spanning tree of $G$ that satisfies the constraints imposed by $D$. The restrictions to add an edge in the tree depend on its…

We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight…

The component size of a graph is the maximum number of edges in any connected component of the graph. Given a graph $G$ and two integers $k$ and $c$, $(k,c)$-Decomposition is the problem of deciding whether $G$ admits an edge partition into…

Given a graph $G=(V,E)$, a tree cover is a collection of trees $\mathcal{T}=\{T_1,T_2,...,T_q\}$, such that for every pair of vertices $u,v\in V$ there is a tree $T\in\mathcal{T}$ that contains a $u-v$ path with a small stretch. If the…

Chv\'{a}tal and Erd\"{o}s [Discrete Math. 2 (1972) 111-113] stated that, for an $m$-connected graph $G$, if its independence number $\alpha(G)\leq m-1$, then $G$ is Hamilton-connected. Note that $k$-leaf-connectedness is a natural…

We prove an ear-decomposition theorem for $4$-edge-connected graphs and use it to prove that for every $4$-edge-connected graph $G$ and every $r\in V(G)$, there is a set of four spanning trees of $G$ with the following property. For every…

We call a tree $T$ is \emph{even} if every pair of its leaves is joined by a path of even length. Jackson and Yoshimoto~[J. Graph Theory, 2024] conjectured that every $r$-regular nonbipartite connected graph $G$ has a spanning even tree.…

We consider the following random model for edge-colored graphs. A graph $G$ on $n$ vertices is fixed, and a random subgraph $G_p$ is chosen by letting each edge of $G$ remain independently with probability $p$. Then, each edge of $G_p$ is…

For any graph $G$, let $t(G)$ be the number of spanning trees of $G$, $L(G)$ be the line graph of $G$ and for any non-negative integer $r$, $S_r(G)$ be the graph obtained from $G$ by replacing each edge $e$ by a path of length $r+1$…

We prove the Strong Nine Dragon Tree Conjecture is true if we replace the edge bound with $d + \big\lceil k \big\lfloor\frac{d-1}{k+1}\big\rfloor \big(\frac{d}{k+1} - \frac{1}{2} \big\lceil\frac{d}{k+1}\big\rceil \big)\big\rceil \leq d +…

In 2012, Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor \frac{3k}{2}\rfloor+m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In 2022,…

A spanning tree of a graph $G$ is a connected acyclic spanning subgraph of $G$. We consider enumeration of spanning trees when $G$ is a $2$-tree, meaning that $G$ is obtained from one edge by iteratively adding a vertex whose neighborhood…

In 1998, Broersma and Tuinstra [J. Graph Theory \textbf{29} (1998), 227-237] proved that if $G$ is a connected graph satisfying $\sigma_2(G) \geq |G|-k+1$ then $G$ has a spanning $k-$ended tree. They also gave an example to show that the…

The Gy\'arf\'as tree packing conjecture asserts that any set of trees with $2,3, ..., k$ vertices has an (edge-disjoint) packing into the complete graph on $k$ vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some special…