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

Let $G$ be a graph (with multiple edges allowed) and let $T$ be a tree in $G$. We say that $T$ is $\textit{even}$ if every leaf of $T$ belongs to the same part of the bipartition of $T$, and that $T$ is $\textit{weakly even}$ if every leaf…

We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of…

In 1970 Lov{\'a}sz gave a necessary and sufficient condition for the existence of a factor $F$ in a graph $G$ such that for each vertex $v$, $g(v)\le d_F(v)\le f(v)$, where $g$ and $f$ are two integer-valued functions on $V(G)$ with $g\le…

In this paper, we estimate the weak saturation numbers of trees. As a case study, we examine caterpillars and obtain several tight estimates. In particular, this implies that for any $\alpha\in [1,2]$, there exist caterpillars with $k$…

In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges.

Let $G$ be a simple graph and let $n,m$ be two integers with $0<m<n$. We prove that $iso(G-S)\leq \frac{n}{m}|S|$ for every $S \subset V(G)$ if and only if $G$ has a $\{C_{2i+1},T \colon 1 \leq i < \frac{m}{n-m},…

Let $a,b,n$ be three positive integers such that $a\equiv b\pmod 2$ and $n\geq b(a+b)(a+b+2)/(2a)$. Let $G$ be a graph of order $n$ with minimum degree at least $a+b/a-1$. We show that $G$ has an $(a,b)$-parity factor, if…

We investigate isometric embeddings of finite metric trees into $(\mathbb{R}^n,d_{1})$ and $( \mathbb{R}^n, d_{\infty})$. We prove that a finite metric tree can be isometrically embedded into $(\mathbb{R}^n,d_{1})$ if and only if the number…

A tree diagram is a tree with positive integral weight on each edge, which is a notion generalized from the Dynkin diagrams of finite-dimensional simple Lie algebras. We introduce two nilpotent Lie algebras and their extended solvable Lie…

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…

We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main result…

A $1-$factorization of a complete graph on $2n$ vertices is said to be $G-$regular if it posseses an automorphism group $G$ acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation…

Let $k$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $\delta$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge…

The Gy\'arf\'as tree packing conjecture states that any set of $n-1$ trees $T_{1},T_{2},..., T_{n-1}$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$. We show that $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$…

The aim of this paper is to provide an affirmative answer to a recent question by Bubeck and Linial on the local profile of trees. For a tree $T$, let $p^{(k)}_1(T)$ be the proportion of paths among all $k$-vertex subtrees (induced…

We show that for every graph $G$ that contains two edge-disjoint spanning trees, we can choose two edge-disjoint spanning trees $T_1,T_2$ of $G$ such that $|d_{T_1}(v)-d_{T_2}(v)|\leq 5$ for all $v \in V(G)$. We also prove the more general…

A digraph $D$ with a subset $S$ of $V(D)$ is called $\boldsymbol{S}${\bf -strong} if for every pair of distinct vertices $u$ and $v$ of $S$, there is a $(u, v)$-dipath and a $(v, u)$-dipath in $D$. We define a digraph $D$ with a subset $S$…

In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as…

A graph $G=(V,E)$ with a vertex set $V$ and an edge set $E$ is called a pairwise compatibility graph (PCG, for short) if there are a tree $T$ whose leaf set is $V$, a non-negative edge weight $w$ in $T$, and two non-negative reals…

A good range of problems on trees can be described by the following general setting: Given a bilinear map $*:\mathbb R^d\times\mathbb R^d\to\mathbb R^d$ and a vector $s\in\mathbb R^d$, we need to estimate the largest possible absolute value…