Imbalance in tournament designs (Q2712527)

A tournament design (TD) is a quadruple \((V,F,P,\alpha)\) where \(V\) is a \(2n\)-element set whose elements are teams, \(F=\{F_1,F_2,\dots,F_{2n-1}\}\) is a set of 1-factorizations such that \((V,F)\) is a 1-factorization of \(K_{2n}\), and \(\alpha =(\alpha_1,\dots,\alpha_{2n-1})\) is a field assignment, i.e. the \(\alpha_i \) maps the \(n\) 2-subsets of \(F_i\) onto the set of fields \(P\). The appearance matrix of a tournament design TD\((n)\) is an \(n \times 2n\) matrix \(A=(a_{ij})\) where the entry \(a_{ij}\) is the number of times the team \(T_j\) plays on the field \(P_i\). This paper introduces two measures of imbalance, team imbalance and field imbalance. Both of these are defined in terms of the appearance matrix. The team imbalance of the team \(T_j\), \(I_T(j)\), is the maximum over \(i\) and \(k\) of \(\{|a_{ij} - a_{kj}|: i,k \in \{1,\dots,n\}\}\), and the total team imbalance \(\text{IT}(D)\) of the tournament design \(D\) is \(\sum_{j=1}^{2n} I_T(j)\). Similarly, the field imbalance of the field \(P_i\), \(I_F(i)\), is the maximum over \(j\) and \(\ell\) of \(\{|a_{ij} - a_{i \ell}|: j,\ell \in \{1,\dots,n\}\}\), and the total field imbalance \(\text{IF}(D)\) of the tournament design \(D\) is \(\sum_{i=1}^{n} I_F(i)\). This paper contains a complete investigation of imbalances in tournament designs with up to eight teams. In addition, the authors present some bounds on the imbalances as well as recursive constructions for homogeneous tournaments.