Duality properties of strong isoperimetric inequalities on a planar graph and combinatorial curvatures

TL;DR

This paper explores the duality of strong isoperimetric inequalities on planar graphs, their relation to combinatorial curvatures, and the connection to Gromov hyperbolicity, providing new theoretical insights and examples.

Contribution

It establishes the equivalence of strong isoperimetric inequalities between planar graphs and their duals under certain conditions, and links these inequalities to Gromov hyperbolicity.

Findings

Duality of isoperimetric inequalities on planar graphs and duals

Strengthening of results relating negative curvature to hyperbolicity

Conditions under which isoperimetric inequalities imply Gromov hyperbolicity

Abstract

This paper is about hyperbolic properties on planar graphs. First, we study the relations among various kinds of strong isoperimetric inequalities on planar graphs and their duals. In particular, we show that a planar graph satisfies a strong isoperimetric inequality if and only if its dual has the same property, if the graph satisfies some minor regularity conditions and we choose an appropriate notion of strong isoperimetric inequalities. Second, we consider planar graphs where negative combinatorial curvatures dominate, and use the outcomes of the first part to strengthen the results of Higuchi, \.{Z}uk, and, especially, Woess. Finally, we study the relations between Gromov hyperbolicity and strong isoperimetric inequalities on planar graphs, and give a proof that a planar graph satisfying a proper kind of a strong isoperimetric inequality must be Gromov hyperbolic if face degrees of the graph are bounded. We also provide some examples to support our results.