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., 2024) in Section 1 (Introduction)

If $G$ is a connected graph on $n$ vertices, then $b(G)\le\lceil\sqrt n\rceil$.

— Exact random covers of metric trees: balanced rounding, duality, and sharp thresholds  (2608.18967 - Wu et al., 19 Aug 2026) in Conjecture 1, Introduction

Theorem~\ref{thm:main} is an exact result for metric trees. It does not by itself prove \cref{conj:burning}. In the transfer from a metric tree to a discrete tree, a center may lie inside an edge, and the discretization of Norin and TurcotteLemma~5.4 enlarges radii. Thus the continuous loss is removed, while the final discrete step remains a separate problem.

— Exact random covers of metric trees: balanced rounding, duality, and sharp thresholds  (2608.18967 - Wu et al., 19 Aug 2026) in Introduction, paragraph immediately following Theorem 1