Related papers: Forbidding Exactly One Hamming Distance

In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and…

We prove that the sign-rank of the $k$-Hamming Distance matrix on $n$ bits is $2^{O(k)}$, independent of the number of bits $n$. This strongly refutes the conjecture of Hatami, Hatami, Pires, Tao, and Zhao (RANDOM 2022), and Hatami,…

A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this…

A graph $\mathcal{H}=(W,E_\mathcal{H})$ is said to have {\em bandwidth} at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_\mathcal{H}$. We say that $\mathcal{H}$ is a…

A random geometric graph, $G(n,r)$, is formed by choosing $n$ points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most $r$. For a given…

For a vertex set $S\subseteq V(G)$ in a graph $G$, the {\em distance multiset}, $D(S)$, is the multiset of pairwise distances between vertices of $S$ in $G$. Two vertex sets are called {\em homometric} if their distance multisets are…

The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove…

We obtain new lower bounds for the independence number of $K_r$-free graphs and linear $k$-uniform hypergraphs in terms of the degree sequence. This answers some old questions raised by Caro and Tuza \cite{CT91}. Our proof technique is an…

Recall that a binary linear code of length $n$ is a linear subspace $\mathcal{C} = \{x\in\mathbb{F}_2^n\mid Ax=0\}$. Here the parity check matrix $A$ is a binary $m\times n$ matrix of rank $m$. We say that $\mathcal{C}$ has rate $R=1-\frac…

Let $A(n, d)$ denote the maximum size of a binary code of length $n$ and minimum Hamming distance $d$. Studying $A(n, d)$, including efforts to determine it as well to derive bounds on $A(n, d)$ for large $n$'s, is one of the most…

Embedding graphs in a geographical or latent space, i.e.\ inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We…

Suppose that $[n]=\left\{0,1,2,...,n\right\}$ is a set of non-negative integers and $h,k \in [n]$. The $L(h,k)$-labeling of graph $G$ is the function $l:V(G)\rightarrow[n]$ such that $\left|l(u)-l(v)\right|\geq h$ if the distance $d(u,v)$…

Let x, y be strings of equal length. The Hamming distance h(x,y) between x and y is the number of positions in which x and y differ. If x is a cyclic shift of y, we say x and y are conjugates. We consider f(x,y), the Hamming distance…

The Hamming graph $H(n,q)$ is defined on the vertex set $[q]^n$ and two vertices are adjacent if and only if they differ in precisely one coordinate. Alon \cite{Alon} proved that the burning number of $H(n,2)$ is $\lceil\frac n2\rceil+1$.…

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$.…

Dvorak (2013) gave a bound on the minimum size of a distance r dominating set in the terms of the maximum size of a distance 2r independent set and generalized coloring numbers, thus obtaining a constant factor approximation algorithm for…

We compute the Hausdorff dimension of the set of simultaneously $q^{-\lambda}$-well approximable points on the Veronese curve in $\mathbb{R}^n$ for $\lambda$ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given…

Let $X_1,..., X_n$ be independent, uniformly random points from $[0,1]^2$. We prove that if we add edges between these points one by one by order of increasing edge length then, with probability tending to 1 as the number of points $n$…

A celebrated theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in $\{0, 1\}^n$ with diameter $d$ has cardinality at most that of a Hamming ball of radius $d/2$. In this paper, we give an algebraic…

In this paper, we study the problem of determining the maximum number of edges in an $n$-vertex $r$-uniform hypergraph that contains no $(k+1)$-connected subgraph. The graph case is a classical problem initiated by Mader, central to graph…