In　2006,　Alon　proposed　a　problem　of　characterizing　all　four-tuples　(n,　m,　s,　d)　such　that　every　digraph　on　n　vertices　of　minimum　out-degree　at　least　s　contains　a　subdigraph　on　m　vertices　of　minimum　out-degree　at　least　d.　He　in　particular　asked　whether　there　exists　an　absolute　constant　c　such　that　every　digraph　on　2n　vertices　of　minimum　out-degree　at　least　s　contains　a　subdigraph　on　n　vertices　of　minimum　out-degree　at　least　2s　-　c?　Recently,　Steiner　resolved　this　case　in　the　negative　by　showing　that　for　arbitrarily　large　n,　there　exists　a　tournament　on　2n　vertices　of　minimum　out-degree　s　=　n　-1,　in　which　the　minimum　outdegree　of　every　subdigraph　on　n　vertices　is　at　most　2s　-　(21　+　o(1))　log3s.　In　this　paper,　we　study　the　above　problem　and　present　two　new　results.　The　first　result　is　that　for　arbitrary　large　nand　any　integer　alpha　＞=　2,　there　exists　a　digraph　on　alpha　n　vertices　of　minimum　out-degree　s　=　n　-1　satisfying　that　the　minimum　out-degree　of　every　subdigraph　on　n　vertices　is　at　most　alpha　s　-　(alpha　1　+o(1))　log　alpha+1　s.　The　second　result　is　that　for　arbitrary　large　nand　any　r　＞=　3,　there　exists　a　digraph　on　2n　vertices　of　girth　rand　minimum　out-degree　s　satisfying　that　the　minimum　out-degree　of　every　subdigraph　on　n　vertices　is　at　most　2　-　(　1　s　2　+o(1))　logrs　if　r　is　odd,　and　is　at　most　2s　-　(21　+o(1))　logr　+1　s　if　r　is　even.　(c)　2024　Elsevier　B.V.　All　rights　are　reserved,　including　those　for　text　and　data　mining,　AI　training,　and　similar　technologies.