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

翻译：图燃烧是图上的离散时间过程，其中顶点被依次激活，且燃烧顶点会随时间推移导致其邻居燃烧。本文研究一种动态场景：图随时间增长，且每一步我们在增长网格$G_n = [-f(n),f(n)]^2$中燃烧顶点。我们考察形如$f(n)=\lceil cn^α\rceil$（其中$α\ge 1$且$c>0$）的函数对应的可达燃烧密度集合。证明表明：当$α=1$时，可达密度集合为$[1/(2c^2),1]$；当$1<α<3/2$时，区间$[0,1]$内所有密度均可达；当$α=3/2$时，可达密度集合为$[0,(1+\sqrt{6}c)^{-2}]$。