Related papers: $K_{r,s}$ graph bootstrap percolation

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square,…

We analyse the jigsaw percolation process, which may be seen as a measure of whether two graphs on the same vertex set are `jointly connected'. Bollob\'as, Riordan, Slivken and Smith proved that when the two graphs are independent binomial…

Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect…

An $r$-uniform hypergraph ($r$-graph) is linear if any two edges intersect at most one vertex. For a graph $F$, a hypergraph $H$ is Berge-$F$ if there is a bijection $\phi:E(F)\rightarrow E(H)$ such that $e\subseteq \phi(e)$ for all $e$ in…

Consider the problem of determining the maximal induced subgraph in a random $d$-regular graph such that its components remain bounded as the size of the graph becomes arbitrarily large. We show, for asymptotically large $d$, that any such…

We study the percolation time of the $r$-neighbour bootstrap percolation model on the discrete torus $(\Z/n\Z)^d$. For $t$ at most a polylog function of $n$ and initial infection probabilities within certain ranges depending on $t$, we…

The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The…

Let $k,l$ be nonnegative integers. A graph $G$ is $(k,l)$-polar if its vertex set admits a partition $(A,B)$ such that $A$ induces a complete multipartite graph with at most $k$ parts, and $B$ induces a disjoint union of at most $l$ cliques…

Hypergraphs are higher-order networks that capture the interactions between two or more nodes. Hypergraphs can always be represented by factor graphs, i.e. bipartite networks between nodes and factor nodes (representing groups of nodes).…

Given nonnegative integers, $s$ and $k$, an $(s,k)$-polar partition of a graph $G$ is a partition $(A,B)$ of $V_G$ such that $G[A]$ and $\overline{G[B]}$ are complete multipartite graphs with at most $s$ and $k$ parts, respectively. If $s$…

We establish the existence of the phase transition in site percolation on pseudo-random $d$-regular graphs. Let $G=(V,E)$ be an $(n,d,\lambda)$-graph, that is, a $d$-regular graph on $n$ vertices in which all eigenvalues of the adjacency…

If each edge (u,v) of a graph G=(V,E) is decorated with a permutation pi_{u,v} of k objects, we say that it has a permuted k-coloring if there is a coloring sigma from V to {1,...,k} such that sigma(v) is different from pi_{u,v}(sigma(u))…

In their seminal paper introducing the theory of random graphs, Erd\H{o}s and R\'{e}nyi considered the evolution of the structure of a random subgraph of $K_n$ as the density increases from $0$ to $1$, identifying two key points in this…

We define a random graph obtained via connecting each point of $\mathbb{Z}^d$ independently to a fixed number $1 \leq k \leq 2d$ of its nearest neighbors via a directed edge. We call this graph the directed $k$-neighbor graph. Two natural…

The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus…

We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $\Delta$ and for $k$-uniform hypergraphs of maximum degree $\Delta$ in which any pair of edges overlaps in at most $r$ vertices. The…

We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards…

Let $K_{s,t}^{(r)}$ denote the $r$-uniform hypergraph obtained from the graph $K_{s,t}$ by inserting $r-2$ new vertices inside each edge of $K_{s,t}$. We prove essentially tight bounds on the size of a largest $K_{s,t}^{(r)}$-subgraph of…

Bootstrap percolation is an often used model to study the spread of diseases, rumors, and information on sparse random graphs. The percolation process demonstrates a critical value such that the graph is either almost completely affected or…

A subset $S$ of a vertex set of a graph $G$ is a total $(k,r)$-dominating set if every vertex $u \in V(G)$ is within distance $k$ of at least $r$ vertices in $S$. The minimum cardinality among all total $(k,r)$-dominating sets of $G$ is…