In a digraph $D=(V,A)$, an oriented path is a sequence $P=x_1x_2\dots x_p$ of distinct vertices such that either $x_ix_{i+1}\in A$ or $x_{i+1}x_{i}\in A$ or both for every $i\in [p-1]$. If $x_ix_{i+1}\in A$ in $P$, then $x_ix_{i+1}$ is a forward arc of $P$; otherwise, $x_{i+1}x_{i}$ is a backward arc. The independence number $α(D)$ is the maximum integer $p$ such that $D$ has a set of $p$ vertices where there is no arc between any pair of vertices. A digraph is $k$-connected if its underlying undirected graph is $k$-connected. Freschi and Lo (JCT-B 2024) proved that every $n$-vertex oriented graph with minimum degree $δ\ge n/2$ has a Hamilton oriented cycle with at most $n-δ$ backward arcs. We prove that every 2-connected digraph $D$ with $α(D)\le 2$ has a Hamilton oriented cycle with at most five backward arcs, and every 1-connected digraph $D$ with $α(D)\le 2$ has a Hamilton oriented path with at most two backward arcs.

翻译：设$D=(V,A)$为一个有向图，定向路径是指顶点序列$P=x_1x_2\dots x_p$，其中顶点互异，且对每个$i\in [p-1]$，要么$x_ix_{i+1}\in A$，要么$x_{i+1}x_{i}\in A$，或二者同时成立。若在$P$中$x_ix_{i+1}\in A$，则$x_ix_{i+1}$为$P$的正向弧；否则，$x_{i+1}x_{i}$为反向弧。独立数$α(D)$是满足$D$中存在$p$个顶点且其中任意两顶点间均无弧的最大整数$p$。有向图称为$k$-连通的，如果其底层的无向图是$k$-连通的。Freschi与Lo (JCT-B 2024) 证明：每个最小度$δ\ge n/2$的$n$顶点定向图都存在一个哈密顿定向环，其中至多包含$n-δ$条反向弧。我们证明：每个满足$α(D)\le 2$的2-连通有向图$D$都存在一个至多包含五条反向弧的哈密顿定向环，且每个满足$α(D)\le 2$的1-连通有向图$D$都存在一个至多包含两条反向弧的哈密顿定向路径。