Improved Bounds for Burning Fence Graphs

Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The burning number of a graph G is the minimum number of rounds necessary for each vertex of G to burn. We consider the burning number of the m × n Cartesian grid graphs, written G m , n . For m = ω ( n ) , the asymptotic value of the burning number of G m , n was determined, but only the growth rate of the burning number was investigated in the case m = O ( n ) , which we refer to as fence graphs. We provide new explicit bounds on the burning number of fence graphs G c n , n , where c > 0 .