The Burning Graph Conjecture for connected graphs

Let $G$ be a connected graph, let $V(G)$ denote its vertex set, and let $b(G)$ be its burning number, the least number of steps needed to burn all vertices of $G$. For a path $P_n$ on $n$ vertices, $b(P_n)=\lceil\sqrt{n}\rceil$.

Burning Graph Conjecture. Every connected graph $G$ satisfies

$$b(G)\leq\lceil\sqrt{|V(G)|}\rceil.$$

Paths attain the bound, so the conjecture asserts that they are extremal for burning number. It remains open in general, although it is known for specific graph classes and the paper establishes asymptotic and bounded-growth results.