Related papers: Visibility in hypercubes

Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority…

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$…

The strong chromatic number, $\chi_S(G)$, of an $n$-vertex graph $G$ is the smallest number $k$ such that after adding $k\lceil n/k\rceil-n$ isolated vertices to $G$ and considering {\bf any} partition of the vertices of the resulting graph…

The mixed metric dimension ${\rm mdim}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that (metrically) resolves each pair of elements from $V(G)\cup E(G)$. We say that $G$ is a max-mdim graph if ${\rm mdim}(G) = n(G)$.…

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $\chi(H)$ and $\chi_l(H)$ be the chromatic number and the…

A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We conjecture that every connected subcubic graph with $m$ edges and $b$ bridges that is distinct from $K_{3,3}$ has a…

The closed neighborhood conflict-free chromatic number of a graph $G$, denoted by $\chi_{CN}(G)$, is the minimum number of colors required to color the vertices of $G$ such that for every vertex, there is a color that appears exactly once…

An $n$-subdivision of a graph $G$ is a graph constructed by replacing a path of length $n$ instead of each edge of $G$ and an $m$-power of $G$ is a graph with the same vertices as $G$ and any two vertices of $G$ at distance at most $m$ are…

Erd\H{o}s asked for the largest size $f(n)$ of a subset of $\{1,\dots,n\}$ with no element dividing two others. We show that $f(n)=c_2\,n+o(n)$ for an effectively computable constant $c_2$, and moreover that the number $q(n)$ of such…

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number $\omega$ has chromatic number at most $3\cdot…

For graphs $G$ and $H$, an $H$-coloring of $G$ is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colorings encode graph theory notions such as independent sets and proper colorings, and are a…

A set $S$ of vertices is a determining set for a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The size of a smallest determining set for $G$ is called its determining number, $Det(G)$. A graph $G$ is…

A vertex $v$ of a 2-connected cubic graph $G$ is $\lambda$-matchable if $G$ has a spanning subgraph in which $v$ has degree three whereas every other vertex has degree one, and we let $\lambda(G)$ denote the number of such vertices.…

A divisor graph $G$ is an ordered pair $(V, E)$ where $V \subset \mathbbm{Z}$ and for all $u \neq v \in V$, $u v \in E$ if and only if $u \mid v$ or $v \mid u$. A graph which is isomorphic to a divisor graph is also called a divisor graph.…

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. To be more formal, a graph $G$ is a median graph if, for all $\mu, u,v\in…

Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1\geq \mu_2\geq \cdots\geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of graph $G$ and let $\sigma=\sigma(G)$ $(1\leq \sigma\leq n)$ be the largest positive integer such…

We introduce the Red-Blue Separation problem on graphs, where we are given a graph $G=(V,E)$ whose vertices are colored either red or blue, and we want to select a (small) subset $S \subseteq V$, called red-blue separating set, such that…

An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves the adjacencies along with their types and directions. The…

For $S \subseteq \mathbb{R}$, positive integer $n$, and $d > 0$, let $G(S^n, d)$ be the graph whose vertex set is $S^n$ where any two vertices are adjacent if and only if they are Euclidean distance $d$ apart. The primary question we will…

In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that…