相关论文: Spectral hypergraph sparsification via chaining

Constructing a sparse spanning subgraph is a fundamental primitive in graph theory. In this paper, we study this problem in the Centralized Local model, where the goal is to decide whether an edge is part of the spanning subgraph by…

In 2012, Ne\v{s}et\v{r}il and Ossona de Mendez proved that graphs of bounded degeneracy that have a path of order $n$ also have an induced path of order $\Omega(\log \log n)$. In this paper we give an almost matching upper bound by…

We study streaming algorithms for the maximum directed cut problem. The edges of an $n$-vertex directed graph arrive one by one in an arbitrary order, and the goal is to estimate the value of the maximum directed cut using a single pass and…

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums of labels taken over the distinct hyperedges are distinct. Let s(H) be the smallest integer N such that there is a sum-distinguishing labeling…

A simple probabilistic argument shows that every $r$-uniform hypergraph with $m$ edges contains an $r$-partite subhypergraph with at least $\frac{r!}{r^r}m$ edges. The celebrated result of Edwards states that in the case of graphs, that is…

In graph sparsification, the goal has almost always been of {global} nature: compress a graph into a smaller subgraph ({sparsifier}) that maintains certain features of the original graph. Algorithms can then run on the sparsifier, which in…

An $r$-uniform hypergraph $H$ is semi-algebraic of complexity $\mathbf{t}=(d,D,m)$ if the vertices of $H$ correspond to points in $\mathbb{R}^{d}$, and the edges of $H$ are determined by the sign-pattern of $m$ degree-$D$ polynomials.…

This paper extends the scenario of the Four Color Theorem in the following way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly) embedded into R^d. We investigate lower and upper bounds on the maximum (weak and…

Sampling edges from a graph in sublinear time is a fundamental problem and a powerful subroutine for designing sublinear-time algorithms. Suppose we have access to the vertices of the graph and know a constant-factor approximation to the…

We present a new algorithm for finding large independent sets in $3$-colorable graphs with small $1$-sided threshold rank. Specifically, given an $n$-vertex $3$-colorable graph whose uniform random walk matrix has at most $r$ eigenvalues…

In 1973, Brown, Erd\H{o}s and S\'os proved that if $\mathcal{H}$ is a 3-uniform hypergraph on $n$ vertices which contains no triangulation of the sphere, then $\mathcal{H}$ has at most $O(n^{5/2})$ edges, and this bound is the best possible…

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…

A graph $G$ is $k$-locally sparse if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ edges. Alon, Krivelevich, and Sudakov showed that for $f > 0$ if a graph $G$ of maximum degree $\Delta$ is…

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Truji\'c…

We derive a new proof to show that the incremental resparsification algorithm proposed by Kelner and Levin (2013) produces a spectral sparsifier in high probability. We rigorously take into account the dependencies across subsequent…

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer {\rm(}which may not be divisible by $k${\rm)} and let $s\le \lfloor n/k\rfloor-1$. We show that if $H$ is a $k$-uniform hypergraph…

The notion of code sparsification was introduced by Khanna, Putterman and Sudan (arxiv.2311.00788), as an analogue to the the more established notion of cut sparsification in graphs and hypergraphs. In particular, for $\alpha\in (0,1)$ an…

We show that for every integer $n\geq 1$ there exists a graph $G_n$ with $(1+o(1))n$ vertices and $n^{1 + o(1)}$ edges such that every $n$-vertex planar graph is isomorphic to a subgraph of $G_n$. The best previous bound on the number of…

We prove that the maximum number of edges in a 3-uniform linear hypergraph on $n$ vertices containing no 2-regular subhypergraph is $n^{1+o(1)}$. This resolves a conjecture of Dellamonica, Haxell, Luczak, Mubayi, Nagle, Person, R\"odl,…

We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding…