Related papers: A tight Erd\H{o}s-P\'osa function for planar minor…

Given a sequence $\mathscr{A}=\{a_0<a_1<a_2\ldots\}\subseteq \mathbb{N}$, let $r_{\mathscr{A},h}(n)$ denote the number of ways $n$ can be written as the sum of $h$ elements of $\mathscr{A}$. Fixing $h\geq 2$, we show that if $f$ is a…

Matching minors are a specialisation of minors fit for the study of graph with perfect matchings. The notion of matching minors has been used to give a structural description of bipartite graphs on which the number of perfect matchings can…

Let H be any graph. We determine (up to an additive constant) the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let delta(H,n) denote the smallest integer t such that…

An immersion of a graph $H$ into a graph $G$ is a one-to-one mapping $f:V(H) \to V(G)$ and a collection of edge-disjoint paths in $G$, one for each edge of $H$, such that the path $P_{uv}$ corresponding to edge $uv$ has endpoints $f(u)$ and…

Let $f=(f^x\mid x\in S)$, $S\subset\mathbb{Z}^m$, be a semigroup of ergodic measure-preserving transformations of a probability space $(\Omega,\mathsf{P})$ and $h$ a real random function on $S$, such that $h(x+y,\omega)\le…

According to the classic Chv{\'{a}}tal's Lemma from 1977, a graph of minimum degree $\delta(G)$ contains every tree on $\delta(G)+1$ vertices. Our main result is the following algorithmic "extension" of Chv\'{a}tal's Lemma: For any…

In this paper we show that there exists a constant $C>0$ such that for any graph $G$ on $Ck\ln k$ vertices either $G$ or its complement $\bar{G}$ has an induced subgraph on $k$ vertices with minimum degree at least $\frac12(k-1)$. This…

Extremal properties of sparse graphs, randomly perturbed by the binomial random graph are considered. It is known that every $n$-vertex graph $G$ contains a complete minor of order $\Omega(n/\alpha(G))$. We prove that adding $\xi n$ random…

We prove that for all $r\geq2$ and c>0, every graph of order n with at least cn^{r} cliques of order r contains a complete r-partite graph with each part of size $\lfloor c^{r}\log n \rfloor.$ This result implies a concise form of the…

The branchwidth of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher''…

For graphs $F$ and $H$, we say $F$ is Ramsey for $H$ if every $2$-coloring of the edges of $F$ contains a monochromatic copy of $H$. The graph $F$ is Ramsey $H$-minimal if $F$ is Ramsey for $H$ and there is no proper subgraph $F'$ of $F$ so…

The orthogonality dimension of a graph $G$ over $\mathbb{R}$ is the smallest integer $k$ for which one can assign a nonzero $k$-dimensional real vector to each vertex of $G$, such that every two adjacent vertices receive orthogonal vectors.…

Graph Sparsification aims at compressing large graphs into smaller ones while preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of…

We consider problems about packing and counting Hamilton $\ell$-cycles in hypergraphs of large minimum degree. Given a hypergraph $\mathcal H$, for a $d$-subset $A\subseteq V(\mathcal H)$, we denote by $d_{\mathcal H}(A)$ the number of…

In their celebrated paper [Ramsey-Type Theorems, Discrete Appl. Math. 25 (1989) 37-52], Erd\H{o}s and Hajnal asked the following: is it true, that for any finite graph H there exists a constant c(H) such that for any finite graph G, if G…

Submodular optimization finds applications in machine learning and data mining. In this paper, we study the problem of maximizing functions of the form $h = f-c$, where $f$ is a monotone, non-negative, weakly submodular set function and $c$…

The Sparsest Cut is a fundamental optimization problem that has been extensively studied. For planar inputs the problem is in $P$ and can be solved in $\tilde{O}(n^3)$ time if all vertex weights are $1$. Despite a significant amount of…

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…

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…

The Planar Graph Metric Compression Problem is to compactly encode the distances among $k$ nodes in a planar graph of size $n$. Two na\"ive solutions are to store the graph using $O(n)$ bits, or to explicitly store the distance matrix with…