Related papers: Pebbling on Jahangir graphs

An obstacle representation of a graph $G$ is a set of points in the plane representing the vertices of $G$, together with a set of polygonal obstacles such that two vertices of $G$ are connected by an edge in $G$ if and only if the line…

The burning number of a graph $G$ is the smallest positive integer $k$ such that the vertex set of $G$ can be covered with balls of radii $0, 1, \dots, k-1$. A well-known conjecture by Bonato, Janssen and Roshabin states that any connected…

In this paper, we introduce the concept of curling subsequence of simple, finite and connected graphs. A curling subsequence is a maximal subsequence $C$ of the degree sequence of a simple connected graph $G$ for which the curling number…

We consider the computational complexity of finding a legal black pebbling of a DAG $G=(V,E)$ with minimum cumulative cost. A black pebbling is a sequence $P_0,\ldots, P_t \subseteq V$ of sets of nodes which must satisfy the following…

The concept of graph burning and burning number ($bn(G)$) of a graph G was introduced recently [1]. Graph burning models the spread of contagion (fire) in a graph in discrete time steps. $bn(G)$ is the minimum time needed to burn a graph…

For $G$ a simple, connected graph, a vertex labeling $f:V(G)\rightarrow \mathbb{Z}_+$ is called a $\textit{radio labeling of}$ $G$ if it satisfies $|f(u)-f(v)|\geq \operatorname{diam}(G) + 1 - d(u,v)$ for all distinct vertices $u,v\in…

Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices. Let $f(n,h) = \Pi_{i}^h a_i$ where…

The blow-up of a graph is obtained by replacing every vertex with a finite collection of copies so that the copies of two vertices are adjacent if and only if the originals are. If every vertex is replaced with the same number of copies,…

Motivated by a graph theoretic process intended to measure the speed of the spread of contagion in a graph, Bonato, Janssen, and Roshanbin [Burning a Graph as a Model of Social Contagion, Lecture Notes in Computer Science 8882 (2014) 13-22]…

Given a graph $G$, a subset $M$ of $V(G)$ is a module of $G$ if for each $v\in V(G)\setminus M$, $v$ is adjacent to all the elements of $M$ or to none of them. For instance, $V(G)$, $\emptyset$ and $\{v\}$ ($v\in V(G)$) are modules of $G$…

In this paper, we study a dynamic coloring of the vertices of a graph $G$ that starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with…

Given a graph G, a subset M of V (G) is a module of G if for each v \in V (G) \diagdownM, v is adjacent to all the elements of M or to none of them. For instance, V(G), \varnothing and {v} (v \in V(G)) are modules of G called trivial. Given…

A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. Let $\{G_1, G_2, \ldots,…

A subset $S$ of initially infected vertices of a graph $G$ is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects…

A \emph{primitive hole} of a graph $G$ is a cycle of length 3 in $G$. The number of primitive holes in a given graph $G$ is called the primitive hole number of the graph $G$. The primitive degree of a vertex $v$ of a given graph $G$ is the…

In the game of \emph{cops and robbers} on a graph $G = (V,E)$, $k$ cops try to catch a robber. On the cop turn, each cop may move to a neighboring vertex or remain in place. On the robber's turn, he moves similarly. The cops win if there is…

A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,\delta)$ denote the minimum number of vertices that need to…

Let $H$ and $G$ be graphs on $n$ vertices, where $n$ is sufficiently large. We prove that if $H$ has Ore-degree at most 5 and $G$ has minimum degree at least $2n/3$ then $H\subset G.$

Graph burning models the spread of information or contagion in a graph. At each time step, two events occur: neighbours of already burned vertices become burned, and a new vertex is chosen to be burned. The big conjecture is known as the…

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 $\lceil\frac n2\rceil+1$.…