Related papers: Buneman's theorem for trees with exatcly n vertice…

Given a set $\Sg$ of bipartitions of some finite set $X$ of cardinality at least 2, one can associate to $\Sg$ a canonical $X$-labeled graph $\B(\Sg)$, called the Buneman graph. This graph has several interesting mathematical properties -…

By weighted tree we understand such connected tree,that: a) each its vertex and each edge have a positive integer weight; b) the weight of each vertex is equal to the sum of weights of outgoing edges. Each tree has a binary structure --- we…

We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $n$ vertices, where $n\geq1$ is a fixed positive integer. The method uses a bijection between mappings…

One of the main aims of phylogenetics is the reconstruction of the correct evolutionary tree when data concerning the underlying species set are given. These data typically come in the form of DNA, RNA or protein alignments, which consist…

We provide the necessary and sufficient conditions for the edge-binomials of the tree forming a $d$-sequence in terms of the degree sequence notion of a graph. We study the regularity of powers of the binomial edge ideals of trees generated…

We prove a density version of the Halpern-L\"{a}uchli Theorem. This settles in the affirmative a conjecture of R. Laver. Specifically, let us say that a tree $T$ is homogeneous if $T$ has a unique root and there exists an integer $b\meg 2$…

A graph $G$ is called a pairwise compatibility graph (PCG, for short) if it admits a tuple $(T,w, d_{\min},d_{\max})$ of a tree $T$ whose leaf set is equal to the vertex set of $G$, a non-negative edge weight $w$, and two non-negative reals…

Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of $d$-ary trees (rooted trees whose vertex outdegrees are bounded from above by $d\geq 2$) with a given number of leaves. We determine the exact…

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.…

Following Poupard's study of strictly ordered binary trees with respect to two parameters, namely, "end of minimal chain" and "parent of maximum leaf" a true Tree Calculus is being developed to solve a partial difference equation system and…

We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree $T$ on $n$ leaves with a path metric $d$, consider the pairwise distances $\{d(u,v)\}$ between leaves. It is well known that these…

A $(1+\varepsilon)$-stretch tree cover of an edge-weighted $n$-vertex graph $G$ is a collection of trees, where every pair of vertices has a $(1+\varepsilon)$-stretch path in one of the trees. The celebrated Dumbbell Theorem by Arya et. al.…

Mubayi and Verstraete conjectured that if $T$ is a tree on $t + 1$ vertices, then any $n$-vertex graph $G$ with average degree $d$ contains at least \[ n d(d - 1) \cdots (d - t + 1) \] labeled copies of $T$ as long as $d$ is sufficiently…

Any algebraic connection on a vector bundle on a smooth complex algebraic curve determines an irregular class and in turn a fission tree at each puncture. The fission trees are the discrete data classifying the admissible deformation…

In this paper, we address the problem of packing large trees in $G_{n,p}$. In particular, we prove the following result. Suppose that $T_1, \dotsc, T_N$ are $n$-vertex trees, each of which has maximum degree at most $(np)^{1/6} / (\log…

Pairwise compatibility graphs (PCGs) with non-negative integer edge weights recently have been used to describe rare evolutionary events and scenarios with horizontal gene transfer. Here we consider the case that vertices are separated by…

We study strong $r$-central $2$-trees whose non-central vertices have degrees in $\{2,3\}$, focusing on the cases $r=1,2,3$. For each $r$, we derive exact degree constraints relating the maximum degree $\Delta$ to the numbers of degree-$3$…

A graph G=(V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers `d' and `D' such that each leaf `u' of T is a node of V and the edge `(u,v) belongs to E' iff `d <= d_T(u, v)…

There is a unique path from the root of a tree to any other vertex. Every vertex, except the root, has a parent: the adjoining vertex on this unique path. This is the conventional definition of the parent vertex. For complete binary trees,…

Menger's Edge Theorem asserts that there exist $k$ pairwise edge-disjoint paths between two vertices in an undirected graph if and only if a deletion of any $k-1$ or less edges does not disconnect these two vertices. Alternatively, there…