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Using polynomials to find lower bounds for $r$-bond bootstrap percolation

Natasha Morrison, Shannon Ogden

Abstract

The $r$-bond bootstrap percolation process on a graph $G$ begins with a set $S$ of infected edges of $G$ (all other edges are healthy). At each step, a healthy edge becomes infected if at least one of its endpoints is incident with at least $r$ infected edges (and it remains infected). If $S$ eventually infects all of $E(G)$, we say $S$ percolates. In this paper we provide recursive formulae for the minimum size of percolating sets in several large families of graphs. We utilise an algebraic method introduced by Hambardzumyan, Hatami, and Qian, and substantially extend and generalise their work.

Using polynomials to find lower bounds for $r$-bond bootstrap percolation

Abstract

The -bond bootstrap percolation process on a graph begins with a set of infected edges of (all other edges are healthy). At each step, a healthy edge becomes infected if at least one of its endpoints is incident with at least infected edges (and it remains infected). If eventually infects all of , we say percolates. In this paper we provide recursive formulae for the minimum size of percolating sets in several large families of graphs. We utilise an algebraic method introduced by Hambardzumyan, Hatami, and Qian, and substantially extend and generalise their work.

Paper Structure

This paper contains 8 sections, 23 theorems, 36 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Products of Trees
  4. Products of Stars
  5. Products of Theta Graphs
  6. Conclusions
  7. Concluding Remarks
  8. Proof of \ref{['Gen Stars exact']}

Key Result

Theorem 1.1

Let $k,r,a_1,...,a_k \in {\mathbb Z}^+$. Let $G_i = S_{a_1}\square \cdots \square S_{a_i}$ for each $i\in [k]$, and let $G_0=K_1$. For $0\leq i\leq \Delta(G_{k-1})$, let $d_i$ denote the number of vertices in $G_{k-1}$ of degree $i$. Then

Figures (4)

  • Figure 1: The vector ${\mathbf p}_v^2$ defined for $v\in V(G)$ with $\deg_G(v)\leq r-2$.
  • Figure 2: The theta graph $H_{6,5}$
  • Figure 3: The $r$-percolating set $F$ for $G\square H_{6,5}$ when $\delta(G)\geq r$. $P_i$ denotes an optimal $i$-percolating set on $G$.
  • Figure 4: The graph $G\square H_{6,5}$ and its edge colouring $c'$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • Proposition 3.1
  • Proposition 3.1
  • ...and 33 more