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Score lists in multipartite hypertournaments. (English) Zbl 1220.05082
Summary: Given nonnegative integers \(n_i\) and \(\alpha_i\) with \(0\leq\alpha_i\leq n_i\) \((i= 1,2,\dots, k)\), an \([\alpha_1,\alpha_2,\dots, \alpha_k]\)-\(k\)-partite hypertournament on \(\sum^k_1 n_i\) vertices is a \((k+1)\)-tuple \((U_1, U_2,\dots, U_k,E)\), where \(U_i\) are \(k\) vertex sets with \(|U_i|= n_i\), and \(E\) is a set of \(\sum^k_1\) \(\alpha_i\)-tuples of vertices, called arcs, with exactly \(\alpha_i\) vertices from \(U_i\), such that any \(\sum^k_1\alpha_i\) subset \(\bigcup^k_1 U_i'\) of \(\bigcup^k_1 U_i\), \(E\) contains exactly one of the \(\left(\sum^k_1\alpha_i\right)!\sum^k_1\) \(\alpha_i\)-tuples whose entries belong to \(\bigcup^k_1 U_i'\). We obtain necessary and sufficient conditions for \(k\) lists of nonnegative integers in nondecreasing order to be the losing score lists and to be the score lists of some \(k\)-partite hypertournament.

05C65 Hypergraphs