Backward Arcs in Hamilton Oriented Cycles and Paths in Directed Graphs with Independence Number Two
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 $\alpha(D)$ 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 $\delta\ge n/2$ has a Hamilton oriented cycle with at most $n-\delta$ backward arcs. We prove that every 2-connected digraph $D$ with $\alpha(D)\le 2$ has a Hamilton oriented cycle with at most five backward arcs, and every 1-connected digraph $D$ with $\alpha(D)\le 2$ has a Hamilton oriented path with at most two backward arcs.