Burning Graph Classes

Abstract

The Burning Number Conjecture, that a graph on n vertices can be burned in at most ⌈ √ n ⌉ 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 nonleaf 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 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).

The Burning Number Conjecture Holds Asymptotically

“…In [2,15], it is used that if there are many leaves in a tree, we can cut them off, burn (obtain a cover of) the remaining subtree and then increment all radii by 1 to burn the entire tree. Here, we have pushed this idea further by using Lemma 2.3 to cut off as much as we need to obtain a subtree with at most 3 remaining leaves.…”

“…Since the introduction of graph burning, this conjecture has been the central open problem in the field. The conjecture is known to hold on multiple classes of graphs, such as spiders [9,8], caterpillars [14], some p-caterpillars [11], sufficiently large graphs with minimum degree at least 4 [2], and others (for instance [15]).…”