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

Let be a connected graph with vertices. Let be an integer. Suppose that a fire breaks out at a vertex of . A firefighter starts to protect vertices. At each step, the firefighter protects -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 denote the maximum number of vertices in that the firefighter can save when a fire breaks out at vertex . The -surviving rate of is defined to be , which is the average proportion of saved vertices. In this paper, we prove that if is a planar graph with vertices and without 5-cycles, then rho(2)(G) > 1/363.