Some criteria for circle packing types and combinatorial Gauss-Bonnet Theorem

Abstract

We investigate criteria for circle packing (CP) types of disk triangulation graphs embedded into simply connected domains in C. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if

Sigma(infinity)(n=1) 1/Sigma(n-1)(j=0) (k(j) + 6) = infinity,

where k(n) is the degree excess sequence defined by

k(n) = Sigma(v)(is an element of Bn) (deg v - 6)

for combinatorial balls B-n of radius n and centered at a fixed vertex. It is also shown that the simple random walk on a disk triangulation graph is recurrent if

Sigma(infinity)(n=1) 1/Sigma(n-1)(j=0) (k(j) + 6) + Sigma(n)(j=0) (k(j) + 6) = infinity.

These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998.