Burning Graph Classes

TL;DR

This paper develops a systematic approach to verify the Burning Number Conjecture for various graph classes, broadening the scope beyond specific cases and providing new results for trees and other graphs.

Contribution

It introduces a general machinery for testing the conjecture across multiple graph classes, extending previous case-by-case methods.

Findings

Proved the conjecture for triangle-free graphs with degree bounds.

Established the conjecture for certain classes of trees with degree and concentration constraints.

Provided a framework applicable to diverse graph classes for verifying the burning number bound.

Abstract

The Burning Number Conjecture, that a graph on $n$ vertices can be burned in at most $\lceil \sqrt{n} \ \rceil$ rounds, has been of central interest for the past several years. Much of the literature toward its resolution focuses on two directions: tightening a general upper bound for the burning number, and proving the conjecture for specific graph classes. In the latter, most of the developments work within a specific graph class and exploit the intricacies particular to it. In this article, we broaden this approach by developing systematic machinery that can be used as test beds for asserting that graph classes satisfy the conjecture. We show how to use these to resolve the conjecture for several classes of graphs including triangle-free graphs with degree lower bounds, graphs with certain linear lower bounds on $r$-neighborhood sizes, all trees whose non-leaf vertices have degree at least $4$, trees whose non-leaf vertices have degree at least $3$ (on at least $81$ vertices), trees whose non-leaf vertices are less than $\frac{2}{3}$ concentrated in degree $2$, and trees with a low concentration of high degree non-leaf vertices (the last two results holding for sufficiently many non-leaf vertices).