  Peter Doyle conjectured that locally univalent circle packings on the
hexagonal lattice only consist of regular hexagonal packings and Doyle spirals,
which is called the Doyle conjecture. In this paper, we prove a rigidity
theorem for Doyle spirals in the class of infinite circle packings on the
hexagonal lattice whose radii ratios of adjacent circles have a uniform bound.
This gives a partial answer to the Doyle conjecture. Based on a new observation
that the logarithmic of the radii ratio of adjacent circles is a weighted
discrete harmonic function, we prove the result via the Liouville theorem of
discrete harmonic functions.
