Links between the Slater Index and the Ryser Index of Tournaments

Abstract.

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 τ˜(T) of T, defined 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 τ˜(T). More precisely, calling I(n, τ) the maximum value of i(T) over the set of tournaments on n vertices and such that τ˜(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.