Links between the Slater Index and the Ryser Index of Tournaments

Given a tournament T, the Slater index i(T] of T is the minimum number of arcs that must be reversed to make T transitive, The Ryser index τ̃ (T) of T, denned from the out-degrees of T, measures a remoteness between T and the transitive tournaments of same order. In this paper, we study some links between i(T) and τ̃(T). More precisely, calling I(n, τ) the maximum value of i(T) over the set of tournaments on n vertices and such that τ̃(T) = τ, we compute an upper bound of I(n, τ) for every value of τ. Then we use this upper bound to study a conjecture stated by J.-C. Bermond on the regularity (i.e., the fact that all the out-degrees are equal or almost equal) of the tournaments with a maximum Slater index by showing that the out-degrees of such tournaments cannot be "too far" from the ones of the regular tournaments.