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 vertexconnectivity 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)+4/4. 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+1/5]. In particular, the conjecture holds for k <= 4 (c) 2020 Elsevier B.V. All rights reserved.