Sparse graphs are not flammable

TL;DR

This paper analyzes the firefighter problem on graphs, establishing thresholds for non-flammability based on edge density, and constructs examples of graphs that are flammable, highlighting the limits of these thresholds.

Contribution

It provides a threshold for the edge density below which graphs are not flammable and constructs flammable graphs to show the bounds are tight.

Findings

Graphs with fewer than ( au_k - psilon)n edges are not flammable.

The surviving rate ar(G) is positive below the threshold.

Constructed random graphs demonstrate the sharpness of the threshold.

Abstract

In this paper, we consider the following \emph{$k$-many firefighter problem} on a finite graph $G=(V,E)$. Suppose that a fire breaks out at a given vertex $v \in V$. In each subsequent time unit, a firefighter protects $k$ vertices which are not yet on fire, and then the fire spreads to all unprotected neighbours of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. The surviving rate $\rho(G)$ of $G$ is defined as the expected percentage of vertices that can be saved when a fire breaks out at a random vertex of $G$. Let $\tau_k = k+2-\frac {1}{k+2}$. We show that for any $\epsilon >0$ and $k \ge 2$, each graph $G$ on $n$ vertices with at most $(\tau_k-\epsilon)n$ edges is not flammable; that is, $\rho(G) > \frac {2\epsilon}{5\tau_k} > 0$. Moreover, a construction of a family of flammable random graphs is proposed to show that the constant $\tau_k$ cannot be improved.