The burning number conjecture holds asymptotically

Abstract

The burning number $b\left(G\right)$ of a graph G is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that $b\left(G\right)\le \lceil \sqrt{n}\rceil$ for all connected graphs G on n vertices. We prove that this conjecture holds asymptotically, that is $b\left(G\right)\le (1+o(1\left)\right)\sqrt{n}$.