The Burning Number Conjecture Holds Asymptotically

TL;DR

This paper proves that the burning number of any graph on n vertices is asymptotically at most (1+o(1)) times the square root of n, confirming the Burning Number Conjecture in the limit.

Contribution

It establishes that the Burning Number Conjecture holds asymptotically, advancing understanding of graph burning dynamics.

Findings

Proves the conjecture asymptotically as n grows large.

Shows that the burning number is at most (1+o(1))√n for large graphs.

Confirms the conjecture's validity in the limit.

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 $b(G)\leq (1+o(1))\sqrt n$.