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