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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.

MSC:
 05C65 Hypergraphs