On the existence of specified cycles in bipartite tournaments

Abstract

For two integers n ≥ 3 and 2 ≤ p ≤ 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 ≥ 3) bipartite tournament BT_4k contains D(4k,p) for all 2 ≤ p ≤ 4k unless BT_4k is isomorphic to a digraph D which has a Hamiltonian cycle (1,2,3, ...,4k, 1), for any vertex i ∊ (1,2,3, ...,4k, 1), there are (4m +1 - l,t) ∊ A(D) and (i,4m + i + 1) ∊ A(D), where 1 ≤ m ≤ k - 1, every vertex i modulo 4k so that the vertex 4k + i is the vertex i. © 2020 Utilitas Mathematica Publishing Inc.. All rights reserved.