Spanning eulerian subdigraphs avoiding k prescribed arcs in tournaments

Fraise and Thomassen (1987) proved that every (k+1)-strong tournament has a hamiltonian cycle which avoids any prescribed set of k arcs. Bang-Jensen, Havet and Yeo showed in Bang-Jensen et al. (2019) that a number of results concerning vertex-connectivity and hamiltonian cycles in tournaments have analogues when we replace vertex connectivity by arc-connectivity and hamiltonian cycles by spanning eulerian subdigraphs. They proved the existence of a smallest function f(k) with the property that every f(k)-arc-strong semicomplete digraph has a spanning eulerian subdigraph which avoids any prescribed set of k arcs by showing that f(k)≤(k+1)2+44. They conjectured that every (k+1)-arc-strong semicomplete digraph has a spanning eulerian subdigraph which avoids any prescribed set of k arcs and verified this for k=2,3. In this paper we prove that f(k)≤⌈6k+15⌉. In particular, the conjecture holds for k≤4