On the number of non-critical vertices in strong tournaments of order with minimum out-degree and in-degree

Let T be a strong tournament of order n ≥ 4 with given minimum out-degree δ + and in-degree δ − . By definition, a vertex w in T is non-critical if the subtournament T − w is also strong. In the present paper, we show that T contains at least min { n , 2 δ + + 2 δ − − 2 } non-critical vertices, and all tournaments for which this lower bound is attained are determined. For the case min { δ + , δ − } ≥ 2 , we also describe all strong tournaments of order n ≥ 2 δ + + 2 δ − that include exactly 2 δ + + 2 δ − − 1 non-critical vertices. From this description it follows that any strong tournament T of order n ≥ 2 δ + + 2 δ − + 2 with min { δ + , δ − } ≥ 2 contains at least 2 δ + + 2 δ − non-critical vertices. Finally, for the case min { δ + , δ − } ≥ 4 , we completely describe all strong tournaments of order n ≥ 2 δ + + 2 δ − + 2 that admit exactly 2 δ + + 2 δ − non-critical vertices. All of these results sharpen those obtained recently by K. Kotani in terms of δ = min { δ + , δ − } .