Related papers: The Tree Stabbing Number is not Monotone

We prove a lower bound on the number of spanning two-forests in a graph, in terms of the number of vertices, edges, and spanning trees. This implies an upper bound on the average cut size of a random two-forest. The main tool is an identity…

The quadratic minimum spanning tree problem and its variations such as the quadratic bottleneck spanning tree problem, the minimum spanning tree problem with conflict pair constraints, and the bottleneck spanning tree problem with conflict…

Given points in Euclidean space of arbitrary dimension, we prove that there exists a spanning tree having no vertices of degree greater than 3 with weight at most 1.559 times the weight of the minimum spanning tree. We also prove that there…

Cayley's formula states that there are $n^{n-2}$ spanning trees in the complete graph on $n$ vertices; it has been proved in more than a dozen different ways over its 150 year history. The complete graphs are a special case of threshold…

We study a question that lies at the intersection of classical research subjects in Topological Graph Theory and Graph Drawing: Computing a drawing of a graph with a prescribed number of crossings on a given set $S$ of points, while…

We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we…

The geometric $\delta$-minimum spanning tree problem ($\delta$-MST) is the problem of finding a minimum spanning tree for a set of points in a normed vector space, such that no vertex in the tree has a degree which exceeds $\delta$, and the…

Let $G=(V,E)$ be a loopless graph and $\mathcal{T}(G)$ be the set of all spanning trees of $G$. Let $L(G)$ be the line graph of the graph $G$ and $t(L(G))$ be the number of spanning trees of $L(G)$. Then, by using techniques from electrical…

We study the minimal spanning arborescence which is the directed analogue of the minimal spanning tree, with a particular focus on its infinite volume limit and its geometric properties. We prove that in a certain large class of transient…

Let $\mathcal T_n$ denote the set of all labelled spanning trees of $K_n$. A family $\mathcal F \subset \mathcal T_n$ is $t$-intersecting if for all $A, B \in \mathcal F$ the trees $A$ and $B$ share at least $t$ edges. In this paper, we…

One theorem of Nemhauser and Trotter ensures that, under certain conditions, a stable set of a graph G can be enlarged to a maximum stable set of this graph. For example, any stable set consisting of only simplicial vertices is contained in…

A metric tree ($M$, $d$), also known as $\mathbb{R}$-trees or $T$-theory, is a metric space such that between any two points there is an unique arc and that arc is isometric to an interval in $\mathbb{R}$. In this paper after presenting…

For a set of graphs $\mathcal{F}$, the extremal number $ex(n;\mathcal{F})$ is the maximum number of edges in a graph of order $n$ not containing any subgraph isomorphic to some graph in $\mathcal{F}$. If $\mathcal{F}$ contains a graph on…

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph…

Let $T$ be a tree, a vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. For two distinct vertices $u,v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v.$ For a…

In the Euclidean Bottleneck Steiner Tree problem, the input consists of a set of $n$ points in $\mathbb{R}^2$ called terminals and a parameter $k$, and the goal is to compute a Steiner tree that spans all the terminals and contains at most…

In the laminar-constrained spanning tree problem, the goal is to find a minimum-cost spanning tree which respects upper bounds on the number of times each cut in a given laminar family is crossed. This generalizes the well-studied…

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\geq 1$, every connected graph on $n$ vertices with minimum degree at least $(\frac{1}{s+3}+o(1))n$ contains a spanning tree having at most $s$ branch…

We prove that if T is a tree on n vertices wih maximum degree D and the edge probability p(n) satisfies: np>c*max{D*logn,n^{\epsilon}} for some constant \epsilon>0, then with high probability the random graph G(n,p) contains a copy of T.…

Let $R$ and $B$ be a set of red points and a set of blue points in the plane, respectively, such that $R\cup B$ is in general position, and let $f:R \to \{2,3,4, \ldots \}$ be a function. We show that if $2\le |B|\le \sum_{x\in R}(f(x)-2) +…