Solution of a Conjecture of Vandell on Decycling Bipartite Tournaments by Deleting Arcs

The decycling index of a digraph is the minimum number of arcs whose removal yields an acyclic digraph. The maximum arc decycling number \(\overline{\nabla}'(m,n)\) is the maximum decycling index among all \(m\times n\) bipartite tournaments. Recently, R.C. Vandell determined the numbers \(\overline{\nabla}'(2,n)\), \(\overline{\nabla}'(3,n)\), and \(\overline{\nabla}'(4,n)\) for all positive integers \(n\), as well as \(\overline{\nabla}'(5,5)\). In this work, we use a computer program to obtain \(\overline{\nabla}'(5,6)\), \(\overline{\nabla}'(6,6)\), and \(\overline{\nabla}'(5,7)\), as well as some results on \(\overline{\nabla}'(6,7)\) and \(\overline{\nabla}'(5,8)\). In particular, \(\overline{\nabla}'(6,6) = 10\), and this confirms a conjecture of Vandell.