Antimagic orientations of even regular graphs

A labeling of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1, \ldots, m\}$. A labeling of $D$ is antimagic if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u\in V(D)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mutze, and Schwartz [On antimagic directed graphs, J Graph Theory 64 (2010) 219--232] initiated the study of antimagic labelings of digraphs, and conjectured that every connected graph admits an antimagic orientation, where an orientation $D$ of a graph $G$ is antimagic if $D$ has an antimagic labeling. It remained unknown whether every disjoint union of cycles admits an antimagic orientation. In this paper, we first answer this question in the positive by proving that every $2$-regular graph has an antimagic orientation. We then show that for any integer $d\ge2$, every connected, $2d$-regular graph has an antimagic orientation. Our technique is new.