On the existence of specified cycles in bipartite tournaments

TL;DR

This paper investigates the existence of specific cycles within bipartite tournaments, establishing conditions under which certain directed cycles are guaranteed or characterized in these regular digraphs.

Contribution

It proves that most regular bipartite tournaments contain all specified cycles, except for a particular structured class of exceptions.

Findings

Most regular bipartite tournaments contain all specified cycles.

Identifies a unique class of bipartite tournaments that do not contain certain cycles.

Provides conditions characterizing when these cycles are absent.

Abstract

For two integers $n\geq 3$ and $2\leq p\leq n$, we denote $D(n,p)$ the digraph obtained from a directed $n$-cycle by changing the orientations of $p-1$ consecutive arcs. In this paper, we show that a family of $k$-regular $(k\geq 3)$ bipartite tournament $BT_{4k}$ contains $D(4k,p)$ for all $2\leq p\leq 4k$ unless $BT_{4k}$ is isomorphic to a digraph $D$ such that $(1,2,3,...,4k,1)$ is a Hamilton cycle and $(4m+i-1,i)\in A(D)$ and $(i,4m+i+1)\in A(D)$, where $1\leq m\leq k-1$.