The 2-surviving rate of planar graphs without 5-cycles

Abstract

Let $G$ be a connected graph with $n\ge 2$ vertices. Let $k\ge 1$ be an integer. Suppose that a fire breaks out at a vertex $v$ of $G$. A firefighter starts to protect vertices. At each step, the firefighter protects $k$-vertices not yet on fire. At the end of each step, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let $\hbox {sn}_k(v)$ denote the maximum number of vertices in $G$ that the firefighter can save when a fire breaks out at vertex $v$. The $k$-surviving rate $\rho _k(G)$ of $G$ is defined to be $\frac{1}{n^2}\sum _{v\in V(G)} {\hbox {sn}}_{k}(v)$, which is the average proportion of saved vertices. In this paper, we prove that if $G$ is a planar graph with $n\ge 2$ vertices and without 5-cycles, then $\rho _2(G)>\frac{1}{363}$.