Graph bootstrap percolation

Abstract

Abstract. Graph bootstrap percolation is a deterministic cellular automaton which was introduced by Bollobás in 1968, and is defined as follows. Given a graph H, and a set G ⊂ E(K n ) of initially 'infected' edges, we infect, at each time step, a new edge e if there is a copy of H in K n such that e is the only not-yet infected edge of H. We say that G percolates in the H-bootstrap process if eventually every edge of K n is infected. The extremal questions for this model, when H is the complete graph K r , were solved (independently) by Alon, Kalai and Frankl almost thirty years ago. In this paper we study the random questions, and determine the critical probability p c (n, K r ) for the K r -process up to a poly-logarithmic factor. In the case r = 4 we prove a stronger result, and determine the threshold for p c (n, K 4 ).

The Zero Forcing Number of Graphs

“…For now, we write k = k − . The proof is identical to the proof of Lemma 3.3 down to equation (8). The case a + b ≥ 24 p remains unchanged (and that is the case which determines k).…”

How to Burn a Graph

“…A simple variation which leads to complex dynamics is to change the rules for nodes to burn. As in graph bootstrap percolation [3], the rules could be varied so nodes burn only if they are adjacent to at least r burned neighbors, where r > 1. We plan on studying this variation in future work.…”

“…, x k ) is called a burning sequence for G. With this notation, the burning number of G is the length of a shortest burning sequence for G; such a burning sequence is referred to as optimal. For example, for the path P 4 with node set {v 1 , v 2 , v 3 , v 4 }, the sequence (v 2 , v 4 ) is an optimal burning sequence; see Figure 1. Note that for a graph G with at least two nodes, we have that b(G) ≥ 2.…”

“…As a simplified, deterministic approach to these questions, we consider a new approach involving a graph process which we call burning. Burning is inspired by graph theoretic processes like Firefighting [5,8,11], graph cleaning [1], and graph bootstrap percolation [3]. There are discrete time-steps or rounds.…”

Saturation in random graphs

“…Note that our lower bound holds as long as G ( n, p ) is weakly K s ‐saturated in K n . This problem was studied by Balogh, Bollobás and Morris , who showed that the probability threshold for G ( n, p ) to be weakly K s ‐saturated is around , where . It might be possible that changes its behavior at the same threshold.…”