The 2-surviving rate of planar graphs without 4-cycles

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 two vertices not yet on fire. At the end of each time interval, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let sn(2)(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 rho(2)(G) of G is defined to be Sigma(v is an element of V(G)) sn(2)(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 without 4-cycles, then rho(2)(G) > 1/76. (C) 2012 Elsevier B.V. All rights reserved.