Improved Bounds for Burning Fence Graphs

TL;DR

This paper investigates the spread of contagion in grid graphs, providing new explicit bounds on the minimum rounds needed for complete burning in fence graphs, especially when one dimension scales with the square root of the other.

Contribution

It introduces improved explicit bounds for the burning number of fence graphs, extending understanding beyond previous asymptotic results.

Findings

Derived new bounds for burning number of fence graphs

Extended analysis to cases where one dimension scales with the square root of the other

Enhanced understanding of contagion spread in grid-like structures

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 $c > 0$.