Improved Bounds for Burning Fence Graphs

Abstract

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 \(m \times n\) Cartesian grid graphs, written \(G_{m,n}\). For \(m = \omega (\sqrt{n})\), 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 \(m = O(\sqrt{n})\), which we refer to as fence graphs. We provide new explicit bounds on the burning number of fence graphs \(G_{c\sqrt{n},n}\), where \(c > 0\).