设g是一个有n个点M条边的连通图.假设火在图g的一条边uV的两个端点燃起,消防员保护若干个没有着火的顶点,火接着蔓延到其他未保护且没有着火的邻点,火和消防员交替地在图g上移动.设Sn(g,uV;(k1,k2))表示当火在边uV的两个端点燃起时,消防员采取第一步保护k1个点,后面每步保护k2个点的策略所能救下的最大顶点数.定义图g的边存活率ρ(g,E;(k1,k2))=∑uV∈E(g)Sn(g,uV;(k1,k2))/nM,即当火随机地在图g的一条边的两个端点燃起时,消防员最多能救下的顶点数的平均率.本文证明了如果g是一个至少有3个点且最小度至少为3的不含4-圈连通平面图,那么ρ(g,E;(4,2))>7/705.Let Gbe a connected graph with n vertices and m edges.Suppose that a fire breaks out at an edge uv of G(two adjacent vertices).At each time interval,the firefighter protects vertices not yet on fire.At the end of each time interval,the fire spreads to all the unprotected vertices that are associated with a neighbour on fire.Then the firefighter and the fire alternately move on the graph.Let sn(G;uv;(k1;k2))denote the maximum number of vertices in Gthat the firefighter can save when a fire breaks out at two adjacent vertices u and v,protecting k1 vertices at the first step and k2 at subsequent steps.The surviving rate of Gdenotesρ(G,e;(k1,k2))=∑uv∈E(G)sn(G,uv;(k1,k2))/nm,which is the average proportion of saved vertices when a fire breaks out at any edge uv.In this paper,we prove that,if Gis a planar graph without 4-cycles andδ(G)≥3,thenρ(G,e;(4,2))>7/705.国家自然科学基金(11171279;11471273); 国家留学基金委项目(201406310108
