  We study minimal graphic functions on complete Riemannian manifolds $\Si$
with non-negative Ricci curvature, Euclidean volume growth and quadratic
curvature decay. We derive global bounds for the gradients for minimal graphic
functions of linear growth only on one side. Then we can obtain a Liouville
type theorem with such growth via splitting for tangent cones of $\Si$ at
infinity. When, in contrast, we do not impose any growth restrictions for
minimal graphic functions, we also obtain a Liouville type theorem under a
certain non-radial Ricci curvature decay condition on $\Si$. In particular, the
borderline for the Ricci curvature decay is sharp by our example in the last
section.
