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

Let $$G$$ G be a connected graph with $$n\ge 2$$ n ≥ 2 vertices. Let $$k\ge 1$$ k ≥ 1 be an integer. Suppose that a fire breaks out at a vertex $$v$$ v of $$G$$ G . A firefighter starts to protect vertices. At each step, the firefighter protects $$k$$ 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 $$\hbox {sn}_k(v)$$ sn k ( v ) denote the maximum number of vertices in $$G$$ G that the firefighter can save when a fire breaks out at vertex $$v$$ v . The $$k$$ k -surviving rate $$\rho _k(G)$$ ρ k ( G ) of $$G$$ G is defined to be $$\frac{1}{n^2}\sum _{v\in V(G)} {\hbox {sn}}_{k}(v)$$ 1 n 2 ∑ v ∈ V ( G ) sn k ( v ) , which is the average proportion of saved vertices. In this paper, we prove that if $$G$$ G is a planar graph with $$n\ge 2$$ n ≥ 2 vertices and without 5-cycles, then $$\rho _2(G)>\frac{1}{363}$$ ρ 2 ( G ) > 1 363 .