Improved pyrotechnics : Closer to the burning graph conjecture

TL;DR

This paper advances understanding of the Burning Number Conjecture by proving asymptotic cases, reducing the problem to finite cases for bounded growth graphs, and improving the upper bound on the burning number.

Contribution

It provides the best-known upper bound on the burning number, reduces the conjecture to finite cases for bounded growth graphs, and extends results to graphs with higher minimum degree.

Findings

Proved the conjecture asymptotically for large graphs with bounded growth.

Established a new upper bound on the burning number: b(G) ≤ √(4n/3) + 1.

Showed the conjecture holds for large graphs with minimum degree at least 4.

Abstract

The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) \le \lceil \sqrt{n} \rceil.$ While the conjecture remains open, we prove that it is asymptotically true when the order of the graph is much larger than its \emph{growth}, which is the maximal distance of a vertex to a well-chosen path in the graph. We prove that the conjecture for graphs of bounded growth reduces to a finite number of cases. We provide the best-known bound on the burning number of a connected graph $G$ of order $n,$ given by $b(G) \le \sqrt{4n/3} + 1,$ improving on the previously known $\sqrt{3n/2}+O(1)$ bound. Using the improved upper bound, we show that the conjecture almost holds for all graphs with minimum degree at least $3$ and holds for all large enough graphs with minimum degree at least $4$. The previous best-known result was for graphs with minimum degree $23$.