Graph Burning On Large $p$-Caterpillars

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.