A note on the surviving rate of 1-planar graphs

For a connected graph G, suppose that a fire breaks out at its vertex and a firefighter starts to protect vertices. At each time interval, the firefighter protects k vertices 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. The k-surviving rate rho k(G) of G is defined to be the expected percentage of vertices saved if the fire breaks out at a random vertex. In this note, we consider the surviving rate of 1-planar graphs, and show that every 1-planar graph G has rho 6(G) > 1/163. (C) 2016 Elsevier B.V. All rights reserved.