A note on antimagic orientations of even regular graphs

Motivated by the conjecture of Hartsfield and Ringel on antimagic labelings of undirected graphs, Hefetz, M\"{u}tze, and Schwartz initiated the study of antimagic labelings of digraphs in 2010. Very recently, it has been conjectured in [Antimagic orientation of even regular graphs, J. Graph Theory, 90 (2019), 46-53.] that every graph admits an antimagtic orientation, which is a strengthening of an earlier conjecture of Hefetz, M\"{u}tze and Schwartz. In this paper, we prove that every $2d$-regular graph (not necessarily connected) admits an antimagic orientation, where $d\ge2$. Together with known results, our main result implies that the above-mentioned conjecture is true for all regular graphs.