Graph burning is a discrete-time process that models the spread of social contagion. Initially, all vertices are unburned. In each round, one unburned vertex is selected and burned, while any unburned vertex that has a burned neighbour from the previous round also becomes burned. The burning number of a graph is the minimum number of rounds needed to burn the entire graph. In this paper, we study the burning number of graph powers. First, we show that for a connected graph $G$, its graph power $G^k$ contains a $(k+1)^+$-branching tree as a spanning tree. A $(k+1)^+$-branching tree is one whose internal vertices have degree at least $k+1$. We then show that $(k+1)^+$-branching trees on $n$ vertices have burning number at most $\left\lceil{\sqrt{\frac{4(k-1)n}{k^2}}}~\right\rceil$. As the burning number of a graph is at most the burning number of any of its spanning trees, this gives an upper bound on the burning number of graph powers. We also derive an explicit bound building on the results of Bastide et al., and identify the ranges of $k$ and $n$ for which our bound outperforms theirs. Finally, we show that $b(G^k) \le (1+o(1))\sqrt{n/k}$ based on the asymptotic burning number bound of Norin and Turcotte.

翻译：图燃烧是一种离散时间过程，用于模拟社会传染的传播。初始状态下所有顶点均未燃烧。在每一轮中，选择一个未燃烧的顶点并将其点燃，同时，任何在前一轮中与已燃烧顶点相邻的未燃烧顶点也将被点燃。图的燃烧数是点燃整个图所需的最少轮数。本文研究图幂的燃烧数。首先，我们证明对于连通图$G$，其图幂$G^k$包含一个$(k+1)^+$分支树作为生成树。$(k+1)^+$分支树是指其内部顶点的度数至少为$k+1$的树。随后，我们证明包含$n$个顶点的$(k+1)^+$分支树的燃烧数至多为$\left\lceil{\sqrt{\frac{4(k-1)n}{k^2}}}~\right\rceil$。由于图的燃烧数不超过其任意生成树的燃烧数，这给出了图幂燃烧数的上界。我们还基于Bastide等人的结果推导出一个显式上界，并确定了我们的上界优于其结果的$k$和$n$取值范围。最后，基于Norin和Turcotte的渐近燃烧数上界，我们证明$b(G^k) \le (1+o(1))\sqrt{n/k}$。