The 2-surviving rate of planar graphs with average degree lower than $4\frac{1}{2}$

Abstract

Let $G$ be any connected graph on $n$ vertices, $n \ge 2.$ Let $k$ be any positive integer. Suppose that a fire breaks out at some vertex of $G.$ Then, in each turn firefighters can protect at most $k$ vertices of $G$ not yet on fire; Next the fire spreads to all unprotected neighbours. The $k$-surviving rate of G, denoted by $\rho_k(G),$ is the expected fraction of vertices that can be saved from the fire, provided that the starting vertex is chosen uniformly at random. In this note, it is shown that for any planar graph $G$ with average degree $4\frac{1}{2} - \epsilon,$ where $\epsilon \in (0, 1],$ we have $\rho_2(G) \ge \frac{2}{9}\epsilon$. In particular, the result implies a significant improvement of the bound for 2-surviving rate for triangle-free planar graphs (Esperet, van den Heuvel, Maffray and Sipma, 2013) and for planar graphs without 4-cycles (Kong, Wang, Zhang, 2012). The proof is done using the separator theorem for planar graphs. This paper is the corrected version of (Gordinowicz, 2018) unified with the corrigendum.