The Burning Number Conjecture Holds Asymptotically

TL;DR

This work resolves the Burning Number Conjecture asymptotically by translating the graph burning problem into a metric-tree framework and developing a probabilistic, fractional covering theory. The authors construct frugal, flexible covers of metric trees and prove a fractional main result that yields near-uniform radii distributions across decomposed pieces, then transfer these results back to graphs via metricization and rounding. The central outcome is the asymptotic bound $b(G)\le (1+o(1))\sqrt{n}$ for connected graphs on $n$ vertices, advancing toward the conjectured exact bound. The methods blend geometric decomposition, probabilistic coverings, and careful rounding, with potential implications for broader radius sequences and future work aiming at removing the asymptotic error term.

Abstract

The burning number $b(G)$ 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 $b(G)\leq \left\lceil\sqrt{n}\right\rceil$ for all graphs $G$ on $n$ vertices. We prove that this conjecture holds asymptotically, that is $b(G)\leq (1+o(1))\sqrt n$.

Theorems & Definitions (39)

• Conjecture 1.1: Burning Number Conjecture
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof