Achievable Burning Densities of Growing Grids

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 $[0,(1+\sqrt{6}c)^{-2}]$.