Achievable Burning Densities of Growing Grids

TL;DR

This paper studies how the density of burning vertices can be achieved in growing grid graphs under a graph burning process, revealing different achievable density ranges depending on the growth rate of the grid.

Contribution

It characterizes the set of achievable burning densities for growing grids with various growth rates, extending understanding of dynamic graph burning processes.

Findings

For linear growth ($eta=1$), achievable densities range from $1/(2c^2)$ to 1.

For growth rates between 1 and 1.5, all densities in [0,1] are achievable.

At growth rate 1.5, achievable densities are in [0, $(1+ oot{6}rom{6}c)^{-2}]$.

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 $f(n)=\lceil cn^\alpha\rceil$, where $\alpha \ge 1$ and $c>0$. We show that for $\alpha=1$, the set of achievable densities is $[1/(2c^2),1]$, for $1<\alpha<3/2$, every density in $[0,1]$ is achievable, and for $\alpha=3/2$, the set of achievable densities is $[0,(1+\sqrt{6}c)^{-2}]$.