  We investigate criteria for circle packing(CP) types of disk triangulation
graphs embedded into simply connected domains in $ \mathbb{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 \[
  \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)} = \infty,
  \] where $k_n$ is the degree excess sequence defined by \[ k_n = \sum_{v \in
B_n} (\mbox{deg}\, v - 6) \] 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 \[
  \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)+\sum_{j=0}^{n} (k_j +6)}
= \infty.
  \] 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.
