Related papers: Stochastic Embeddings of Graphs into Trees

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

In the Densest k-Subgraph problem, given a graph G and a parameter k, one needs to find a subgraph of G induced on k vertices that contains the largest number of edges. There is a significant gap between the best known upper and lower…

For a graph $G$, let $a(G)$ denote the maximum size of a subset of vertices that induces a forest. We prove the following. 1. Let $G$ be a graph of order $n$, maximum degree $\Delta>0$ and maximum clique size $\omega$. Then \[ a(G) \geq…

The main paradigm of smoothed analysis on graphs suggests that for any large graph $G$ in a certain class of graphs, perturbing slightly the edges of $G$ at random (usually adding few random edges to $G$) typically results in a graph having…

We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence $\boldsymbol{d}=(d_1,\ldots, d_n)$, provided that the number of edges is at least $n + \textstyle{\frac{1}{2}}…

Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is…

For any set $\Omega$ of non-negative integers such that $\{0,1\}\subseteq \Omega$ and $\{0,1\}\ne \Omega$, we consider a random $\Omega$-$k$-tree ${\sf G}_{n,k}$ that is uniformly selected from all connected $k$-trees of $(n+k)$ vertices…

Local convergence of bounded degree graphs was introduced by Benjamini and Schramm. This result was extended further by Lyons to bounded average degree graphs. In this paper we study the convergence of random tree sequences with given…

We prove that for all $0\leq t\leq k$ and $d\geq 2k$, every graph $G$ with treewidth at most $k$ has a `large' induced subgraph $H$, where $H$ has treewidth at most $t$ and every vertex in $H$ has degree at most $d$ in $G$. The order of $H$…

Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R^k. The primary use of this embedding has been for practical spectral clustering algorithms [SM00,NJW02]. Recently,…

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…

The famous Erd\H{o}s-S\'os conjecture states that every graph of average degree more than $t-1$ must contain every tree on $t+1$ vertices. In this paper, we study a spectral version of this conjecture. For $n>k$, let $S_{n,k}$ be the join…

We show that every n-point tree metric admits a (1+eps)-embedding into a C(eps) log n-dimensional L_1 space, for every eps > 0, where C(eps) = O((1/eps)^4 log(1/eps)). This matches the natural volume lower bound up to a factor depending…

In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond percolation is applied.…

We introduce a dense counterpart of graph degeneracy, which extends the recently-proposed invariant symmetric difference. We say that a graph has sd-degeneracy (for symmetric-difference degeneracy) at most $d$ if it admits an elimination…

We show that a randomly perturbed digraph, where we start with a dense digraph $D_\alpha$ and add a small number of random edges to it, will typically contain a fixed orientation of a bounded degree spanning tree. This answers a question…

A famous conjecture of Erd\H{o}s and S\'os states that every graph with average degree more than $k - 1$ contains all trees with $k$ edges as subgraphs. We prove that the Erd\H{o}s-S\'os conjecture holds approximately, if the size of the…

Let $\mathcal{C}$ and $\mathcal{D}$ be hereditary graph classes. Consider the following problem: given a graph $G\in\mathcal{D}$, find a largest, in terms of the number of vertices, induced subgraph of $G$ that belongs to $\mathcal{C}$. We…

We prove that for any finite tree $T$ with $n$ vertices and maximal degree $3$, there is a topological embedding of $T$ into the integer grid $Z^2$ which maps vertices to vertices and whose image meets at most $\frac{7}{3}n$ vertices. This…

An added edge to a graph is called an inset edge. Predicting k inset edges which minimize the average distance of a graph is known to be NP-Hard. When k = 1 the complexity of the problem is polynomial. In this paper, we further find the…