Related papers: Upper bounds on the running time of bootstrap perc…

This paper focuses on the time constant for last passage percolation on complete graph. Let $G_n=([n],E_n)$ be the complete graph on vertex set $[n]=\{1,2,\ldots,n\}$, and i.i.d. sequence $\{X_e:e\in E_n\}$ be the passage times of edges.…

We consider the $r$-neighbor bootstrap percolation process on the graph with vertex set $V=\{0,1\}^n$ and edges connecting the pairs at Hamming distance $1,2,\dots,k$, where $k\ge 2$. We find asymptotics of the critical probability of…

The Hamming torus of dimension $d$ is the graph with vertices $\{1,\dots,n\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $\theta$ starts with a random set of open…

Given two $3$-graphs $F$ and $H$, an $F$-covering of $H$ is a collection of copies of $F$ in $H$ such that each vertex of $H$ is contained in at least one copy of them. Let {$c_2(n,F)$} be the maximum integer $t$ such that every 3-graph…

In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain…

Let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-set $[n]=\{1,2,\ldots,n\}$. Let $n>2k$ and let $\mathcal{F}\subset \binom{[n]}{k}$ be an {\it intersecting} $k$-graph, i.e., $F\cap F'\neq \emptyset$ for all…

We investigate the behaviour of $r$-neighbourhood bootstrap percolation on the binomial $k$-uniform random hypergraph $H_k(n,p)$ for given integers $k\geq 2$ and $r\geq 2$. In $r$-neighbourhood bootstrap percolation, infection spreads…

Given a hypergraph $\mathcal{H}$, the $\mathcal{H}$-bootstrap process starts with an initial set of infected vertices of $\mathcal{H}$ and, at each step, a healthy vertex $v$ becomes infected if there exists a hyperedge of $\mathcal{H}$ in…

For $r\geq1$, the $r$-neighbour bootstrap process in a graph $G$ starts with a set of infected vertices and, in each time step, every vertex with at least $r$ infected neighbours becomes infected. The initial infection percolates if every…

Majority bootstrap percolation on a graph $G$ is an epidemic process defined in the following manner. Firstly, an initially infected set of vertices is selected. Then step by step the vertices that have more infected than non-infected…

Given graphs H_1,...,H_k, we study the minimum order of a graph G such that for each i, the induced copies of H_i in G cover V(G). We prove a general upper bound of twice the sum of the numbers m_i, where m_i is one less than the order of…

A graph $G$ is contractible to a graph $H$ if there is a set $X \subseteq E(G)$, such that $G/X$ is isomorphic to $H$. Here, $G/X$ is the graph obtained from $G$ by contracting all the edges in $X$. For a family of graphs $\cal F$, the…

In 2013, Orlin proved that the max flow problem could be solved in $O(nm)$ time. His algorithm ran in $O(nm + m^{1.94})$ time, which was the fastest for graphs with fewer than $n^{1.06}$ arcs. If the graph was not sufficiently sparse, the…

The bootstrap percolation (or threshold model) is a dynamic process modelling the propagation of an epidemic on a graph, where inactive vertices become active if their number of active neighbours reach some threshold. We study an…

Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The…

We consider first-passage percolation on the class of "high-dimensional" graphs that can be written as an iterated Cartesian product $G\square G \square \dots \square G$ of some base graph $G$ as the number of factors tends to infinity. We…

Given a directed graph $G = (V, E)$ with $n$ vertices, $m$ edges and a designated source vertex $s\in V$, we consider the question of finding a sparse subgraph $H$ of $G$ that preserves the flow from $s$ up to a given threshold $\lambda$…

For graphs $F$ and $H$, let $f_{F,H}(n)$ be the minimum possible size of a maximum $F$-free induced subgraph in an $n$-vertex $H$-free graph. This notion generalizes the Ramsey function and the Erd\H{o}s--Rogers function. Establishing a…

A bootstrap percolation process on a graph G is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round every uninfected node which has at least r infected neighbours…

We study the maximum flow problem in directed H-minor-free graphs where H can be drawn in the plane with one crossing. If a structural decomposition of the graph as a clique-sum of planar graphs and graphs of constant complexity is given,…