A lower bound of the surviving rate of a planar graph with girth at least seven

Abstract

Let G be a connected graph with n≥2 vertices. Suppose that a fire breaks out at a vertex v of G. A firefighter starts to protect vertices. At each time interval, the firefighter protects one vertex not yet on fire. At the end of each time interval, the fire spreads to all the unprotected vertices that have a neighbor on fire. Let sn(v) denote the maximum number of vertices in G that the firefighter can save when a fire breaks out at vertex v. The surviving rate ρ(G) of G is defined to be ∑ _v∈V(G)sn(v)/n ², which is the average proportion of saved vertices.

In this paper, we show that if G is a planar graph with n≥2 vertices and having girth at least 7, then $\rho(G)>\frac{1}{301}$.