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

TL;DR

This paper establishes a lower bound on the 2-surviving rate in planar graphs with average degree below 4.5, improving previous bounds for specific subclasses like triangle-free and cycle-free planar graphs using separator theorems.

Contribution

It provides a new lower bound on the 2-surviving rate for planar graphs with average degree less than 4.5, extending and improving earlier results for special graph classes.

Findings

Lower bound of 2/9 * epsilon on the 2-surviving rate for certain planar graphs.

Improves bounds for triangle-free planar graphs.

Enhances understanding of fire spread resistance in planar graphs.

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.