The edge surviving rate of Halin graphs
Xiaoxue Hu, Wenting Guo, Youmin Qi, Jiangxu Kong · Utilitas Mathematica · 2025
Let \(k\ge 1\) be an integer. Let \(G=(V,E)\) be a connected graph with \(n\) vertices and \(m\) edges. Suppose fires break out at two adjacent vertices. In each round, a firefighter can protect \(k\) vertices, and then the fires spread to all unprotected neighbors. For \(uv\in E(G)\), let \(sn_{k}(uv)\) denote the maximum number of vertices the firefighter can save when fires break out at the ends of \(uv\). The \(k\)-edge surviving rate \(\rho'_k(G)\) of \(G\) is defined as the average proportion of vertices saved when the starting vertices of the fires are chosen uniformly at random over all eages, i.e., \(\rho'_k(G)=\sum\limits_{uv\in E(G)}sn_{k}(uv)/nm\). In particular, we write \(\rho'(G)=\rho'_1(G)\). For a given class of graphs \(\mathcal{G}\) and a constant \(\varepsilon>0\), we seek the minimum value \(k\) such that \(\rho'_k(G)>\varepsilon\) for all \(G\in \mathcal{G}\). In this paper, we prove that for Halin graphs, this minimum value is exactly 1. Specifically, every Halin graph \(G\) satisfies \(\rho'(G)> 1/12\).