TWMS J. Pure Appl. Math., V.5, N.2, 2014, pp.185-193 DEGREE LISTS OF K-BIPARTITE HYPERTOURNAMENTS S. PIRZADA 1 , ZHOU GUOFEI 2 Abstract. In this paper, we obtain necessary and sufficient conditions for a pair of non- decreasing sequences of non-negative integers to be the degree lists of some k-bipartite hyper- tournament. Keywords: hypertournaments, bipartite hypertournaments, degree lists AMS Subject Classification: 05C65 1. Introduction Given two positive integers n and k with n>k> 1, a k-hypertournament H on n vertices is a pair (V,A), where V is a set of n vertices and A is a set of k-tuples of vertices, called arcs, such that for any k-subset W of V , A contains exactly one of the k! possible k-tuples whose entries belong to W . Clearly, a 2-hypertournament is a tournament. If e =(x 1 ,x 2 , ··· ,x k ), then {x 1 ,x 2 , ··· ,x k } is called the underlying vertex set of e, and is denoted by V e . A k- hypertournament H (V,A) is said to be transitive if we can label V (H ) by v 1 , v 2 , ..., v n in such a way that: if 1 ≤ i 1 <i 2 < ··· <i k ≤ n, then (v i 1 ,v i 2 , ··· ,v i k ) is an arc in H . Bipartite hypergraphs are generalization of bipartite graphs. If U = {u 1 ,u 2 , ··· ,u m } and V = {v 1 ,v 2 , ··· ,v n } are vertex sets, then each edge of a bipartite hypergraph is a subset of the vertex sets, containing at least one vertex from U and at least one vertex from V . If each edge has exactly k vertices, the bipartite hypergraph is called a k-bipartite hypergraph. A k-bipartite hypertournament is a complete k-bipartite hypergraph with each edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the hyperedge. A k-bipartite hypertournament is said to be transitive, if it is obtained from a transitive k- hypertournament by deleting all the edges contained in U or V . An oriented k-hypergraph is a k-hypergraph with each edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the hyperedge. Let a =(x 1 ,x 2 ,...,x k ) be an arc of an oriented k-hypergraph H . We call x i the i-th entry of a; the (i + 1)-th entry of a, x i+1 , is called the successor of x i , and x i the predecessor of x i+1 in a,1 ≤ i ≤ k − 1. It is obvious that x k has no successor, and x 1 has no predecessor in a. Define a function ρ on a by ρ(a, x)=  k − i, if x ∈ a and x is the i-th entry of a 0, if x ∈ a For v ∈ V (H ), we denote d + H (v)= ∑ a∈H ρ(a, v) (or simply d + (v)) the degree of v in H . 1 Department of Mathematics, University of Kashmir,Srinagar, Kashmir, India e-mail:pirzadasd@kashmiruniversity.ac.in 2 Department of Mathematics, Nanjing University,Nanjing, P.R. China e-mail: gfzhou@mail.nju.edu.cn Manuscript received March 2014. 185