  Given a digraph $D$ with $m$ arcs and a bijection $\tau: A(D)\rightarrow \{1,
2, \ldots, m\}$, we say $(D, \tau)$ is an antimagic orientation of a graph $G$
if $D$ is an orientation of $G$ and no two vertices in $D$ have the same
vertex-sum under $\tau$, where the vertex-sum of a vertex $u$ in $D$ under
$\tau$ is the sum of labels of all arcs entering $u$ minus the sum of labels of
all arcs leaving $u$. Hefetz, M\"{u}tze, and Schwartz in 2010 initiated the
study of antimagic orientations of graphs, and conjectured that every connected
graph admits an antimagic orientation. This conjecture seems hard, and few
related results are known. However, it has been verified to be true for regular
graphs, biregular bipartite graphs, and graphs with large maximum degree. In
this paper, we establish more evidence for the aforementioned conjecture by
studying antimagic orientations of graphs $G$ with independence number at least
$|V(G)|/2$ or at most four. We obtain several results. The method we develop in
this paper may shed some light on attacking the aforementioned conjecture.
