Related papers: On Packing Low-Diameter Spanning Trees

We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result…

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…

We consider the problem of packing edge-disjoint Steiner forests in a graph. The input consists of a multi-graph $G=(V,E)$ and a collection of $h$ vertex subsets $S = \{S_1,S_2,\ldots,S_h\}$. A Steiner forest for $S$, also called an…

Let $\tau(G)$ and $\kappa'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with…

The tree breadth ${\rm tb}(G)$ of a connected graph $G$ is the smallest non-negative integer $\rho$ such that $G$ has a tree decomposition whose bags all have radius at most $\rho$. We show that, given a connected graph $G$ of order $n$ and…

Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have…

Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be…

Chung and Graham [J. London Math. Soc., 1983] claimed that there exists an $n$-vertex graph $G$ containing all $n$-vertex trees as subgraphs that has at most $\frac{5}{2}n \log_2 n + O(n)$ edges. We identify an error in their proof. This…

Given a graph $G(V,E)$, a vertex subset $S$ of $G$ is called an open packing in $G$ if no pair of distinct vertices in $S$ have a common neighbour in $G$. The size of a largest open packing in $G$ is called the open packing number,…

We introduce the notion of \emph{bounded diameter arboricity}. Specifically, the \emph{diameter-$d$ arboricity} of a graph is the minimum number $k$ such that the edges of the graph can be partitioned into $k$ forests each of whose…

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$…

The Barat-Thomassen conjecture, recently proved in [Bensmail et al.: A proof of the Barat-Thomassen conjecture. J. Combin. Theory Ser. B, 124:39-55, 2017.], asserts that for every tree T, there is a constant $c_T$ such that every $c_T$-edge…

In this paper, we show that every $O(m)$-edge-connected simple graph $G$ of size divisible by $m$ with minimum degree at least $2^{O(m)}$ has an edge-decomposition into isomorphic copies of any given tree $T$ of size $m$. Moreover, the…

We prove a refinement of the tree packing theorem by Tutte/Nash-Williams for finite graphs. This result is used to obtain a similar result for end faithful spanning tree packings in certain infinite graphs and consequently to establish a…

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph $GK_n$ on any set $S$ of $n$ points in general position in the plane? We show that this number is in…

We study the arboricity A and the maximum number T of edge-disjoint spanning trees of the Erdos-Renyi random graph G(n,p). For all p(n) in [0,1], we show that, with high probability, T is precisely the minimum between delta and…

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on…

The longstanding Nash-Williams conjecture asserts that every $K_3$-divisible graph $G$ with $\delta(G)\ge 3n/4$ admits a triangle decomposition. In the random setting, Frankl and R\"odl showed that, with high probability, $G(n,p)$ contains…

Let $s(n)$ be the minimum number of edges in a graph that contains every $n$-vertex tree as a subgraph. Chung and Graham [J. London Math. Soc. 1983] claim to prove that $s(n)\leqslant O(n\log n)$. We point out a mistake in their proof. The…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…