Graph burning is a graph process that models the spread of social contagion. Initially, all the vertices of a graph $G$ are unburnt. At each step, an unburnt vertex is put on fire and the fire from burnt vertices of the previous step spreads to their adjacent unburnt vertices. This process continues till all the vertices are burnt. The burning number $b(G)$ of the graph $G$ is the minimum number of steps required to burn all the vertices in the graph. The burning number conjecture by Bonato et al. states that for a connected graph $G$ of order $n$, its burning number $b(G) \leq \lceil \sqrt{n} \rceil$. It is easy to observe that in order to burn a graph it is enough to burn its spanning tree. Hence it suffices to prove that for any tree $T$ of order $n$, its burning number $b(T) \leq \lceil \sqrt{n} \rceil$ where $T$ is the spanning tree of $G$. It was proved in 2018 that $b(T) \leq \lceil \sqrt{n + n_2 + 1/4} +1/2 \rceil$ for a tree $T$ where $n_2$ is the number of degree $2$ vertices in $T$. In this paper, we provide an algorithm to burn a tree and we improve the existing bound using this algorithm. We prove that $b(T)\leq \lceil \sqrt{n + n_2 + 8}\rceil -1$ which is an improved bound for $n\geq 50$. We also provide an algorithm to burn some subclasses of the binary tree and prove the burning number conjecture for the same.

翻译：图燃烧是一种模拟社会传染扩散的图过程。初始时，图$G$的所有顶点均未被点燃。每一步中，选择一个未被点燃的顶点使其燃烧，同时上一步中已燃烧顶点会将火势蔓延至相邻的未被点燃顶点。此过程持续进行直至所有顶点都被点燃。图$G$的燃烧数$b(G)$是点燃图中所有顶点所需的最少步骤数。Bonato等人提出的燃烧数猜想指出：对于阶数为$n$的连通图$G$，其燃烧数$b(G) \leq \lceil \sqrt{n} \rceil$。容易观察到，要燃烧一个图，只需燃烧其生成树即可。因此，只需证明对于任意阶数为$n$的树$T$（其中$T$是$G$的生成树），其燃烧数$b(T) \leq \lceil \sqrt{n} \rceil$。2018年已证明，对于树$T$，有$b(T) \leq \lceil \sqrt{n + n_2 + 1/4} +1/2 \rceil$，其中$n_2$是$T$中度数为$2$的顶点数。本文提出了一种燃烧树的算法，并利用该算法改进了现有界。我们证明$b(T)\leq \lceil \sqrt{n + n_2 + 8}\rceil -1$，这是对$n\geq 50$情况的改进界。此外，我们还给出了燃烧二叉树某些子类的算法，并验证了针对这些子类的燃烧数猜想。