Backward Arcs in Hamilton Oriented Cycles and Paths in Directed Graphs with Independence Number Two

Abstract

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.

Figures (3)

• Figure 1: An example when $k=m=3$
• Figure 2: Case 2
• Figure 3: An example when $k=6$. The dashed lines indicate $T$ and $Q_6$ are fully connected.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Lemma 2.2