Restricted arc-connectivity of bipartite tournaments

In [L. Volkmann, Restricted arc-connectivity of digraphs, Inform. Process. Lett. 103 (2007) 234–239], L. Volkmann introduced a concept of restricted arc-connectivity   for a digraph DD, where the size of a minimum restricted arc-cut is denoted by λ′(D)λ′(D). The restricted arc-connectivity offers a more refined parameter than the arc-connectivity to measure the reliability of networks. For the investigation of λ′(D)λ′(D) the minimum arc-degree  ξ′(D)ξ′(D) is a useful parameter, introduced by S. Wang and S. Lin in [S. Wang, S. Lin, λ′λ′-optimal digraphs, Inform. Process. Lett. 108 (2008) 386–389].In this paper, we study the restricted arc-connectivity of bipartite tournaments and show that λ′(T)≤ξ′(T)λ′(T)≤ξ′(T) for all strong bipartite tournaments except for a family TT, where λ′(T)=2>1=ξ′(T)λ′(T)=2>1=ξ′(T) for each T∈TT∈T. Furthermore, we prove that all strong bipartite tournaments with δ(T)≥(n+3)/8δ(T)≥(n+3)/8 are optimally restricted arc-connected  , i.e. λ′(T)=ξ′(T)λ′(T)=ξ′(T), and present examples to show the sharpness of this result.