Can every connected graph burn in $\lceil \sqrt{n} \rceil $ steps? While this conjecture remains open, we prove that it is asymptotically true when the graph is much larger than its \emph{growth}, which is the maximal distance of a vertex to a well-chosen path in the graph. In fact, we prove that the conjecture for graphs of bounded growth boils down to a finite number of cases. Through an improved (but still weaker) bound for all trees, we argue that the conjecture almost holds for all graphs with minimum degree at least $3$ and holds for all large enough graphs with minimum degree at least $4$. The previous best lower bound was $23$.

翻译：连接的图形能以 $\ lceil\ sqrt{n}\ rcele $ 步骤刻录吗? 虽然这个猜测仍然开放, 但是我们证明, 当图形大于其 \ emph{ greeng} 时, 它几乎是非抽象的, 也就是顶点与图中选取的路径的最大距离。 事实上, 我们证明, 约束增长的图形的推测会归结为数量有限的案例。 通过改进( 但仍然弱点), 约束所有树, 我们论证说, 通过改进( 但仍然弱点), 将所有至少 $3 的图形都保存在最小水平上, 并且将所有足够大的图形都保留在最低水平上, 至少 $ 4 。 上一个最低的框是 23 美元 。