On the existence of specified cycles in bipartite tournaments

Abstract

For two integers $n\ge 3$ and $2\le p\le n$, 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 $(k\ge 3)$ bipartite tournament $B{T}_{4k}$ contains $D(4k,p)$ for all $2\le p\le 4k$ unless $B{T}_{4k}$ 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 $1\le m\le k-1$.

15 pages