Spanning k-arc-strong subdigraphs with few arcs in k-arc-strong tournaments

Given a k-arc-strong tournament T , we estimate the minimum number of arcs possible in a k-arc-strong spanning subdigraph of T . We give a construction which shows that for each k ≥ 2 there are tournaments T on n vertices such that every k-arc-strong spanning subdigraph of T contains at least nk+ k(k−1) 2 arcs. In fact, the tournaments in our construction have the property that every spanning subdigraph with minimum inand out-degree at least k has nk + k(k−1) 2 arcs. This is best possible since it can be shown that every k-arc-strong tournament contains a spanning subdigraph with minimum inand out-degree at least k and no more than nk + k(k−1) 2 arcs. As our main result we prove that every k-arcstrong tournament contains a spanning k-arc-strong subdigraph with no more than nk + 280k2 arcs. We conjecture that for every k-arc-strong tournament T , the minimum number of arcs in a k-arc-strong spanning subdigraph of T is equal to the minimum number of arcs in a spanning subdigraph of T with the property that every vertex has inand out-degree at least k. We also discuss the implications of our results on related problems and conjectures.