标题： 图燃烧在大型$p$-毛虫上 显示英文标题

标题： Graph Burning On Large $p$-Caterpillars

摘要： 图燃烧模拟了图中信息或传染的传播。 在每个时间步骤中，发生两个事件：已经燃烧的顶点的邻居变得燃烧，选择一个新的顶点进行燃烧。 这个大猜想被称为{\it 燃烧数猜想}：对于任何具有$n$个顶点的连通图，所有$n$个顶点可以在最多$\lceil \sqrt{n}\ \rceil$个时间步骤内被燃烧。 众所周知，为了证明这个猜想，只需证明对于树结构成立。 我们证明了对于足够大的$p$-毛虫树，该猜想成立。 显示英文摘要

摘要： Graph burning models the spread of information or contagion in a graph. At each time step, two events occur: neighbours of already burned vertices become burned, and a new vertex is chosen to be burned. The big conjecture is known as the {\it burning number conjecture}: for any connected graph on $n$ vertices, all $n$ vertices can be burned after at most $\lceil \sqrt{n}\ \rceil$ time steps. It is well-known that to prove the conjecture, it suffices to prove it for trees. We prove the conjecture for sufficiently large $p$-caterpillars.