Improved pyrotechnics : Closer to the burning graph conjecture

Abstract

Can every connected graph burn in ⌈ √ n⌉ steps? While this conjecture remains open, we prove that it is asymptotically true when the graph is much larger than its growth, which is the maximal distance of a vertex to a well-chosen path in the graph. In fact, we prove that the conjecture for graphs of bounded growth boils down to a finite number of cases. Through an improved (but still weaker) bound for all trees, we argue 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 lower bound was 23.How fast can a rumor propagate in a graph? One measure of that, introduced by Bonato, Janssen and Roshanbin [BJR16], is the burning number b(G) of a graph G. At step 1, we set a vertex on fire. At every step i 2, all neighbours of a vertex on fire catch fire themselves, and we set a new vertex on fire. If at the end of step k the whole graph is on fire, then the graph is k-burnable. The burning number b(G) of G is defined to be the least k such that G is k-burnable.A graph with n isolated vertices is trivially not (n − 1)-burnable. We focus on connected graphs. Paths are an interesting special case. For a path P n on n vertices, it is not hard to check that b(P n ) = ⌈ √ n⌉. When introducing the notion, Bonato et al.[BJR16] conjectured that paths are, essentially, the worst case for the burning number of a graph. Conjecture 1.1 (Bonato et al. [BJR16]). Every connected graph G satisfies b(G) ⌈ |V (G)|⌉.Conjecture 1.1 is only known to hold with a constant factor. For any connected graph G and any spanning tree T of G, we have b(G) b(T ). Therefore, it is sufficient to prove that the conjecture holds for trees. While Conjecture 1.1 is still open in general, it has been shown to hold for specific graph classes. Kamali, Miller and Zhang showed [KMZ20] that the conjecture holds for graphs of minimum degree at least 23. Here, we observe that the conjecture almost holds for all connected graphs of minimum degree at least 3, and fully holds for those of minimum degree at least 4 that are large enough.

Burning Graph Classes

“…For the first part, this follows from the given inequality and squaring inequality (8). For the second part, observe our given inequality is satisfied when (k − 3)p k + (k − 2)p k+1 + • • • > 1 3 and this in turn is satisfied when the stronger inequality (k − 3)(p k + p k+1 + • • • ) > 1 3 is satisfied. The result follows.…”

“…This result was improved significantly for connected graphs on sufficiently many vertices in a recent article [1].…”

“…and we search for m n,d . Notice g n,d (x) is constant on [1,2), and for such x the difference g n,d (x) −…”

“…Then G ′ has n ′ many vertices. The graph G ′ is itself connected so by [1] we can burn G ′ in at most…”

“…A graph that satisfies the Burning Number Conjecture is said to be well-burnable, and though the conjecture has not been resolved, the state of the art bound of 4n 3 + O(1) was recently established by Bonato et al [6] and independently Bastide et al [1]. Alongside improving the general upper bound for b(G), a common theme amongst papers in the graph burning literature is proving that specific graph classes are well-burnable.…”