Achievable Burning Densities of Growing Grids

Abstract

Graph burning is a discrete-time process on graphs where vertices are sequentially activated and burning vertices cause their neighbours to burn over time. In this work, we focus on a dynamic setting in which the graph grows over time, and at each step we burn vertices in the growing grid $G_n = [-f(n),f(n)]^2$. We investigate the set of achievable burning densities for functions of the form $f(n)=\lceil cn^α\rceil$, where $α\ge 1$ and $c>0$. We show that for $α=1$, the set of achievable densities is $[1/(2c^2),1]$, for $1<α<3/2$, every density in $[0,1]$ is achievable, and for $α=3/2$, the set of achievable densities is $[0,(1+\sqrt{6}c)^{-2}]$.

Figures (3)

• Figure 1: The first three turns of a burning process with $f(1)=f(2)=1$ and $f(3)=3$. Vertex $(1,1)$ is activated on turn 1 and no other vertices are activated on subsequent turns. The burned vertices are highlighted as red squares.
• Figure 2: A near-cover of a $w$ by $\ell$ rectangle with diamond tiles centred at points $(i, j)$ for $0 \leq i \leq w$, $0 \leq j \leq \ell$ such that $i+j \equiv 1 \mod 2$. Note that smooth-edged diamonds are drawn for simplicity. Each diamond is in fact a collection of burning squares forming a diamond with stair-case edges.
• Figure 3: The gray center square represents $G_n$ and the whole square represents $G_{n+k}$. Each of the four labelled rectangles have two sides of length $w(n,k) = 2c n^{3/2}$ and two sides of length $\ell(n,k) = c \left( (n+k)^{3/2} - n^{3/2} \right)$.

Theorems & Definitions (34)

• Theorem : BGS20
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof