Light structures in infinite planar graphs without the strong isoperimetric property

It is shown that the discharging method can be successfully applied on infinite planar graphs of subexponential growth and even on those graphs that do not satisfy the strong edge isoperimetric inequality. The general outline of the method is presented and the following applications are given: Planar graphs with only finitely many vertices of degree $\le 5$ and with subexponential growth contain arbitrarily large finite submaps of the tessellation of the plane or of some tessellation of the cylinder by equilateral triangles. Every planar graph with isoperimetric number zero and with essential minimum degree $\ge3$ has infinitely many edges whose degree sum is at most 15. In particular, this holds for all graphs with minimum degree $\ge3$ and with subexponential growth. The cases without infinitely many edges whose degree sum is $\le14$ (or, similarly, $\le13$ or $\le 12$) are also considered. Several further results are obtained.