Related papers: Minors, connectivity, and diameter in randomly per…

We prove that for every complete multipartite graph $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as ${n\choose 2}-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into…

We consider the following question. We have a dense regular graph $G$ with degree $\alpha n$, where $\alpha>0$ is a constant. We add $m=o(n^2)$ random edges. The edges of the augmented graph $G(m)$ are given independent edge weights $X(e)$,…

The Bandwidth theorem of B\"ottcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ that ensures $G$ contains every $r$-chromatic graph $H$ on $n$ vertices of bounded degree and of bandwidth $o(n)$,…

The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this…

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a…

We study an inhomogeneous sparse random graph on [N] = {1, . . . , N } as introduced in a seminal paper by Bollobas, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices…

The Graph Minors Structure Theorem of Robertson and Seymour asserts that, for every graph $H,$ every $H$-minor-free graph can be obtained by clique-sums of ``almost embeddable'' graphs. Here a graph is ``almost embeddable'' if it can be…

A randomly perturbed graph $G^p = G_\alpha \cup G(n,p)$ is obtained by taking a deterministic $n$-vertex graph $G_\alpha = (V, E)$ with minimum degree $\delta(G)\geq \alpha n$ and adding the edges of the binomial random graph $G(n,p)$…

This paper concerns the large deviations of a system of interacting particles on a random graph. There is no stochasticity, and the only sources of disorder are the random graph connections, and the initial condition. The average number of…

Given a graph $G$ and a constant $\gamma \in [0,1]$, let $\omega^{(\gamma)}(G)$ be the largest integer $r$ such that there exists an $r$-vertex subgraph of $G$ containing at least $\gamma \binom{r}{2}$ edges. It was recently shown that…

We study Ramsey properties of randomly perturbed $3$-uniform hypergraphs. For~$t\geq 2$, write $\tilde K^{(3)}_t$ to denote the $3$-uniform {\it expanded} clique hypergraph obtained from the complete graph $K_t$ by expanding each of the…

A graph $H$ is said to be an induced minor of a graph $G$ if $H$ can be obtained from $G$ by a sequence of vertex deletions and edge contractions. Equivalently, $H$ is an induced minor of $G$ if there exists an induced minor model of $H$ in…

A seminal result by Koml\'os, Sark\"ozy, and Szemer\'edi states that if a graph $G$ with $n$ vertices has minimum degree at least $kn/(k + 1)$, for some $k \in \mathbb{N}$ and $n$ sufficiently large, then it contains the $k$-th power of a…

A classical result by Otter shows that the complete graph has an exponential number of non-isomorphic spanning trees. This was recently extended by Lee to every almost regular graph of sufficiently large degree. In this paper, we consider…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

Given graphs $F$ and $G$, a perfect $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$ that together cover all the vertices in $G$. The study of the minimum degree threshold forcing a perfect $F$-tiling in a graph $G$…

For an edge-ordered graph $G$, we say that an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with any new label to $H$ introduces a copy of $G$. The saturation function describes the minimum…

We present near-optimal algorithms for detecting small vertex cuts in the CONGEST model of distributed computing. Despite extensive research in this area, our understanding of the vertex connectivity of a graph is still incomplete,…

We study the following question raised by Erd\H{o}s and Hajnal in the early 90's. Over all $n$-vertex graphs $G$ what is the smallest possible value of $m$ for which any $m$ vertices of $G$ contain both a clique and an independent set of…

We study a variety of natural constructions from topological combinatorics, including matching complexes as well as other graph complexes, from the perspective of the graph minor category of \parencite{MiProRa}. We prove that these…