Sparse graphs are not flammable

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 $ρ(G)$ 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 $τ_k = k+2-\frac {1}{k+2}$. We show that for any $ε>0$ and $k \ge 2$, each graph $G$ on $n$ vertices with at most $(τ_k-ε)n$ edges is not flammable; that is, $ρ(G) > \frac {2ε}{5τ_k} > 0$. Moreover, a construction of a family of flammable random graphs is proposed to show that the constant $τ_k$ cannot be improved.

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