Independence numbers of hypergraphs with sparse neighborhoods

AbstractLet H be a hypergraph with N vertices and average degree d. Suppose that the neighborhoods of H are sparse, then its independence number is at least cN(logd /d), where c>0 is a constant. In particular, let integers r≥3 and n≥1 be fixed, and let H be r-uniform, triangle-free and linear, then its independence number is at least cNlognd/d for all sufficiently large d

Note on the degree sequences of k-hypertournaments

AbstractWe obtain a criterion for determining whether or not a non-decreasing sequence of non-negative integers is a degree sequence of some k-hypertournament on n vertices. This result generalizes the corresponding theorems on tournaments proposed by Landau [H.G. Landau, On dominance relations and the structure of animal societies. III. The condition for a score structure, Bull. Math. Biophys. 15 (1953) 143–148] in 1953

On Score Sequences ofk-Hypertournaments

AbstractGiven two nonnegative integers n and k withn≥k> 1, a k -hypertournament on n vertices is a pair (V, A), where V is a set of vertices with | V | =n and A is a set of k -tuples of vertices, called arcs, such that for any k -subset S ofV , A contains exactly one of the k!k -tuples whose entries belong to S. We show that a nondecreasing sequence (r1, r2,⋯ , rn) of nonnegative integers is a losing score sequence of a k -hypertournament if and only if for each j(1 ≤j≤n),with equality holding whenj=n. We also show that a nondecreasing sequence (s1,s2 ,⋯ , sn) of nonnegative integers is a score sequence of somek -hypertournament if and only if for each j(1 ≤j≤n),with equality holding whenj=n. Furthermore, we obtain a necessary and sufficient condition for a score sequence of a strong k -hypertournament. The above results generalize the corresponding theorems on tournaments

Score lists in multipartite hypertournaments

Given non-negative integers $n_{i}$ and $\alpha_{i}$ with $0 \leq \alpha_{i} \leq n_i$ $(i=1,2,...,k)$, an $[\alpha_{1},\alpha_{2},...,\alpha_{k}]$-$k$-partite hypertournament on $\sum_{1}^{k}n_{i}$ vertices is a $(k+1)$-tuple $(U_{1},U_{2},...,U_{k},E)$, where $U_{i}$ are $k$ vertex sets with $|U_{i}|=n_{i}$, and $E$ is a set of $\sum_{1}^{k}\alpha_{i}$-tuples of vertices, called arcs, with exactly $\alpha_{i}$ vertices from $U_{i}$, such that any $\sum_{1}^{k}\alpha_{i}$ subset $\cup_{1}^{k}U_{i}^{\prime}$ of $\cup_{1}^{k}U_{i}$, $E$ contains exactly one of the $(\sum_{1}^{k} \alpha_{i})!$ $\sum_{1}^{k}\alpha_{i}$-tuples whose entries belong to $\cup_{1}^{k}U_{i}^{\prime}$. We obtain necessary and sufficient conditions for $k$ lists of non-negative integers in non-decreasing order to be the losing score lists and to be the score lists of some $k$-partite hypertournament

On Scores, Losing Scores and Total Scores in Hypertournaments

A $k$-hypertournament is a complete $k$-hypergraph with each $k$-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a $k$-hypertournament, the score $s_{i}$ (losing score $r_{i}$) of a vertex $v_{i}$ is the number of arcs containing $v_{i}$ in which $v_{i}$ is not the last element (in which $v_{i}$ is the last element). The total score of $v_{i}$ is defined as $t_{i}=s_{i}-r_{i}$. In this paper we obtain stronger inequalities for the quantities $\sum_{i\in I}r_{i}$, $\sum_{i\in I}s_{i}$ and $\sum_{i\in I}t_{i}$, where $I\subseteq \{ 1,2,\ldots,n\}$. Furthermore, we discuss the case of equality for these inequalities. We also characterize total score sequences of strong $k$-hypertournaments