Related papers: Unweighted One-Sided Code Sparsifiers and Thin Sub…

A weighted (directed) graph is a (directed) graph with integer weights assigned to its vertices and edges. The weight of a subgraph is the sum of weights of vertices and edges in the subgraph. The problem of determining the largest order…

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…

Given a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ if $V(G)\subset V(\mathcal{H})$ and there is a bijection $f:E(G)\rightarrow E(\mathcal{H})$ such that for any edge $e$ of $G$ we have $e\subset f(e)$. We study Ramsey…

We study sequences of linear or affine codes with uniform weight spectrum, i.e., a part of codewords with any fixed weight tends to zero. It is proved that a sequence of linear codes has a uniform weight spectrum if the number of vectors…

Let n be an even positive integer and F be the field \GF(2). A word in F^n is called balanced if its Hamming weight is n/2. A subset C \subseteq F^n$ is called a balancing set if for every word y \in F^n there is a word x \in C such that y…

In this study, linear codes having their Lee-weight distributions over the semi-local ring $\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^{2}=1$ are constructed using the defining set and Gauss sums for an odd prime $q $. Moreover, we derive…

We consider the problem of constructing deletion correcting codes over a binary alphabet and take a graph theoretic view. An $n$-bit $s$-deletion correcting code is an independent set in a particular graph. We propose constructing such a…

Given a graph $F$ and a positive integer $n$, the weak $F$-saturation number $\mathrm{wsat}(K_n,F)$ is the minimum number of edges in a graph $H$ on $n$ vertices such that the edges missing in $H$ can be added, one at a time, so that every…

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time…

A graph $G$ is $(c,t)$-sparse if for every pair of vertex subsets $A,B\subset V(G)$ with $|A|,|B|\geq t$, $e(A,B)\leq (1-c)|A||B|$. In this paper we prove that for every $c>0$ and integer $\ell$, there exists $C>1$ such that if an…

This paper presents efficient distributed algorithms for a number of fundamental problems in the area of graph sparsification: We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in…

A linear graph code is a family $\mathcal{C}$ of graphs on $n$ vertices with the property that the symmetric difference of the edge sets of any two graphs in $\mathcal{C}$ is also the edge set of a graph in $\mathcal{C}$. In this article,…

A code is a subset of the vertex set of a Hamming graph. The set of $s$-neighbours of a code is the set of all vertices at Hamming distance $s$ from their nearest codeword. A code $C$ is $s$-elusive if there exists a distinct code $C'$ that…

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle $C_5$. We show that every graph of order $n$ and size $\left( 1-\frac{1}{k}\right)\binom{n}{2}$, where $k\ge 3$…

For an undirected $n$-vertex graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a minimum $st$-cut in $G$? We solve this problem in preprocessing…

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 celebrated Erd\H{o}s-P\'{o}sa Theorem, in one formulation, asserts that for every $c\geq 1$, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of $c$ cycles have bounded treewidth. What can we say about…

The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic…

We prove that for any integer $k\geq 2$ and $\varepsilon>0$, there is an integer $\ell_0\geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+\varepsilon)n$ has a fractional decomposition into…

Let $G = (V,E)$ be a graph, and for each $e \in E(G)$, let $L_e$ be a list of real numbers. Let $w:E(G) \to \cup_{e \in E(G)}L_e$ be an edge weighting function such that $w(e) \in L_e$ for each $e \in E(G)$, and let $c_w$ be the vertex…