Let $D$ be a digraph. A closed anti-directed Euler trail of $D$ is a closed trail in which consecutive arcs have opposite directions and each arc of $D$ occurs exactly once. A digraph is anti-Eulerian if it contains a closed anti-directed Euler trail. A tournament $T$ is anti-Eulerian if and only if both $d^{+}\left(v\right)$ and $d^{-}\left(v\right)$ are even for any $v\in V\left(T\right)$. The Cartesian product of two directed cycles $\overrightarrow{C_{n_1}}\square \overrightarrow{C_{n_2}}$ is anti-Eulerian if and only if $gcd(n_1,n_2)=1$. If $n_1,n_2,\dots ,n_{2k}$ can be partitioned into $k$ relatively prime pairs, then $\overrightarrow{C_{n_1}}\square \overrightarrow{C_{n_2}}\square \cdots \square \overrightarrow{C_{n_{2k}}}$ is anti-Eulerian. If each of $D_i$ ($1\le i\le k$) is an anti-Eulerian digraph and at least one of them is not a bipartite digraph, then $D_1\square D_2\square \cdots \square D_k$ is anti-Eulerian.

让 $D$是一个有向图。一个封闭的反方向欧拉轨迹$D$ 是一条封闭的轨迹，其中连续的弧具有相反的方向，每个弧的 $D$恰好发生一次。如果一个有向图包含一个封闭的反定向欧拉轨迹，那么它就是反欧拉轨迹。一场比赛$吨$ 是反欧拉的当且仅当两者 $d^{+}\left(v\right)$ 和 $d^{-}\left(v\right)$ 甚至对于任何 $v\in 伏\left(吨\right)$. 两个有向循环的笛卡尔积$\overrightarrow{C_{n_1}}\square \overrightarrow{C_{n_2}}$ 是反欧拉当且仅当 $GCd(n_1,n_2)=1$. 如果$n_1,n_2,\dots ,n_{2克}$ 可以分为 $克$ 相对质数对，那么 $\overrightarrow{C_{n_1}}\square \overrightarrow{C_{n_2}}\square \cdots \square \overrightarrow{C_{n_{2克}}}$是反欧拉的。如果每个$D_{\mathrm{一世}}$ ($1\le \mathrm{一世}\le 克$) 是一个反欧拉有向图并且其中至少一个不是二部有向图，那么 $D_1\square D_2\square \cdots \square D_{克}$ 是反欧拉的。