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 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 article, we give 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. Also, under certain restriction of degree $2$ vertices, we have improved upon the result of Bonato et al.(in 2021). We also provide an algorithm to burn a 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传播的图过程。初始时，图G的所有顶点均未燃烧。每一步中，将一个未燃烧的顶点点燃，且上一轮燃烧的顶点会将其火焰传播到相邻的未燃烧顶点。此过程持续直至所有顶点均被燃烧。图G的燃烧数$b(G)$是燃烧图中所有顶点所需的最少步数。Bonato等人提出的燃烧数猜想指出：对于阶数为n的连通图G，其燃烧数满足$b(G) \leq \lceil \sqrt{n} \rceil$。容易观察到，燃烧一个图只需燃烧其生成树即可。因此，只需证明对任意阶数为n的树T（其中T为G的生成树），其燃烧数满足$b(T) \leq \lceil \sqrt{n} \rceil$。2018年已证明对树T有$b(T) \leq \lceil \sqrt{n + n_2 + 1/4} + 1/2 \rceil$，其中$n_2$是T中度为2的顶点数。本文给出一种燃烧树的算法，并利用该算法改进了现有界，证明$b(T) \leq \lceil \sqrt{n + n_2 + 8}\rceil -1$，此为改进后的界。此外，在对度为2的顶点施加特定限制的条件下，我们进一步改进了Bonato等人（2021年）的结果。我们还提供了一种燃烧二叉树的算法，并针对该类树证明了燃烧数猜想。