Improved Pyrotechnics: Closer to the Burning Number Conjecture

The Burning Number Conjecture claims that for every connected graph G of order n, its burning number satisfies b(G) <= [root n]. While the conjecture remains open, we prove that it is asymptotically true when the order of the graph is much larger than its 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) <= root 4n/3 + 1, improving on the previously known root 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.