Burning number sqrt(n) bound conjecture

Prove that for every connected graph G on n vertices, the burning number b(G)—the minimum number of rounds needed in the burning process where each round selects one new burning source and burning spreads to neighbors—is at most ⌈√n⌉.

Background

The burning number b(G) models fast contagion: in each round, a new source is ignited and burning spreads to neighbors; the process ends when all vertices are burning. For connected graphs of order n, the best general upper bound currently known is b(G) ≤ √(4n/3) + 1, while paths achieve b(P_n) = ⌈√n⌉.

The conjecture asks to tighten the universal bound to match the path case for all connected graphs, and is a central open problem in the burning literature referenced in the introduction.

References

For a connected graph $G$ of order $n$, it is known that $b(G) \le \sqrt{4n/3} + 1,$ and it is conjectured that the bound can be improved to $\lceil\sqrt{n}\rceil$ (which is the burning number of a path of order $n$).

— How to cool a graph (2401.03496 - Bonato et al., 7 Jan 2024) in Section 1 (Introduction)