Related papers: Boolean dimension and dim-boundedness: Planar cove…

Given a poset $P$, a family $F$ of elements in the Boolean lattice is said to be $P$-saturated if (1) $F$ contains no copy of $P$ as a subposet and (2) every proper superset of $F$ contains a copy of $P$ as a subposet. The maximum size of a…

The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for…

Let $\mathcal{C}$ be a class of graphs that is closed under taking subgraphs. We prove that if for some fixed $0<\delta\le 1$, every $n$-vertex graph of $\mathcal{C}$ has a balanced separator of order $O(n^{1-\delta})$, then any depth-$k$…

A labeling scheme for nearest common ancestors assigns a distinct binary string, called the label, to every node of a tree, so that given the labels of two nodes (and no further information about the topology of the tree) we can compute the…

The metric dimension of a graph is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. Bailey and Meagher obtained an upper bound on the…

Schnyder characterized planar graphs in terms of order dimension. Brightwell and Trotter proved that the dimension of the vertex-edge-face poset $\Pvef{M}$ of a planar map $M$ is at most four. In this paper we investigate cases where…

Let $La(n,P)$ be the maximum size of a family of subsets of $[n]=\{1,2,...,n\}$ not containing $P$ as a (weak) subposet. The diamond poset, denoted $B_{2}$, is defined on four elements $x,y,z,w$ with the relations $x<y,z$ and $y,z<w$.…

For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a…

We prove two theorems concerning incidence posets of graphs, cover graphs of posets and a related graph parameter. First, answering a question of Haxell, we show that the chromatic number of a graph is not bounded in terms of the dimension…

Lawson and Osserman proved that the Dirichlet problem for the minimal surface system is not always solvable in the class of Lipschitz maps. However, it is known that minimizing sequences (for area) of Lipschitz graphs converge to objects…

Consider a graph $G$ drawn on a fixed surface, and assign to each vertex a list of colors of size at least two if $G$ is triangle-free and at least three otherwise. We prove that we can give each vertex a color from its list so that each…

In this paper, we prove that the zero-divisor graph $\Gamma(P)$ of a Boolean poset $P$ is both well-covered and Cohen--Macaulay. Furthermore, for a poset $\mathbf{P} = \prod_{i=1}^{n} P_i$ $(n \ge 3)$, where each $P_i$ is a finite bounded…

Filled arrays of bolometers are currently being employed for use in astronomy from the far-infrared through millimeter parts of the electromagnetic spectrum. Because of the large range of wavelengths for which such detectors are applicable,…

The orthogonality dimension of a graph $G=(V,E)$ over a field $\mathbb{F}$ is the smallest integer $t$ for which there exists an assignment of a vector $u_v \in \mathbb{F}^t$ with $\langle u_v,u_v \rangle \neq 0$ to every vertex $v \in V$,…

We show that a set of small box counting dimension can be covered by a H\"older graph from all but a small set of directions, and give sharp bounds for the dimension of the exceptional set, improving a result of B. Hunt and V. Kaloshin. We…

Let $m$ be a positive integer, $X$ a graph with vertex set $\Omega$, and ${\rm WL}_m(X)$ the coloring of the Cartesian $m$-power $\Omega^m$, obtained by the $m$-dimensional Weisfeiler-Leman algorithm. The ${\rm WL}$-dimension of the graph…

The metric dimension of a graph $G$ is the size of a smallest subset $L \subseteq V(G)$ such that for any $x,y \in V(G)$ with $x\not= y$ there is a $z \in L$ such that the graph distance between $x$ and $z$ differs from the graph distance…

The metric dimension of a graph is the smallest number of nodes required to identify all other nodes based on shortest path distances uniquely. Applications of metric dimension include discovering the source of a spread in a network,…

A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist real numbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of the linear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristic…

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In…