The structure of strong $k$-quasi-transitive digraphs with large diameters

TL;DR

This paper characterizes the structure of large-diameter, strong $k$-quasi-transitive digraphs, revealing that their subgraphs are either semicomplete, bipartite, or empty, depending on the parity of $k$.

Contribution

It extends previous results by describing the structure of $k$-quasi-transitive digraphs for odd $k eq 4$, identifying specific subgraph types based on $k$'s parity.

Findings

For odd $k eq 4$, $D[V(P)]$ is semicomplete or semicomplete bipartite.

For odd $k eq 4$, $D[V(D)ackslash V(P)]$ is semicomplete, bipartite, or empty.

The structure depends on the parity of $k$, generalizing earlier even-$k$ results.

Abstract

Let $k$ be an integer with $k\geq 2$. A digraph $D$ is $k$-quasi-transitive, if for any path $x_0x_1\ldots x_k$ of length $k$, $x_0$ and $x_k$ are adjacent. Suppose that there exists a path of length at least $k+2$ in $D$. Let $P$ be a shortest path of length $k+2$ in $D$. Wang and Zhang [Hamiltonian paths in $k$-quasi-transitive digraphs, Discrete Mathematics, 339(8) (2016) 2094--2099] proved that if $k$ is even and $k\ge 4$, then $D[V(P)]$ and $D[V(D)\setminus V(P)]$ are both semicomplete digraphs. In this paper, we shall prove that if $k$ is odd and $k\ge 5$, then $D[V(P)]$ is either a semicomplete digraph or a semicomplete bipartite digraph and $D[V(D)\setminus V(P)]$ is either a semicomplete digraph, a semicomplete bipartite digraph or an empty digraph.