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Let G be an infinite locally finite plane graph with one end and let H be a finite plane subgraph of G . Denote by a(H) the number of finite faces of H and by l(H) the number of the edges of H that are on the boundary of the infinite face or a finite face not in H . Define the isoperimetric constant h (G) to be inf H l(H) / a(H) and define the isoperimetric constant h (¿) to be inf G h (G) where the infimum is taken over all infinite locally finite plane graphs G having minimum degree ¿ and exactly one end. We establish the following bounds on h (¿) for ¿ ¿ 7 : $$ {{(\\delta -6)(\\delta^2 -8 \\delta + 15)}\\over {(\\delta-4)(\\delta^2 - 8\\delta + 13)}} \\le h (\\delta) \\le \\sqrt {{\\delta-6}\\over {\\delta-2}}. $$

Isoperimetric Constants of Infinite Plane Graphs