The Edge-Isoperimetric Problem For Regular Planar Tesselations

Solutions for the edge-isoperimetric problem on the graphs of the triangular and hexagonal tesselations of the Euclidean plane are given. The proofs are based on the fact that their symmetry group is Coxeter. In each case there is certain nice quotient of the stability order of the graph (which is itself a quotient of the Bruhat order of the Coxeter group by a parabolic subgroup).