Improved pyrotechnics : Closer to the burning graph conjecture
share
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.
READ FULL TEXT