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Motzkin and Straus established a close connection between the maximum clique problem and a solution (namely graph-Lagrangians) to the maximum value of a class of homogeneous quadratic multilinear functions over the standard simplex of the Euclidean space in 1965. This connection provides a new proof of Turans theorem. Recently, an extension of Motzkin-Straus theorem was proved for non-uniform hypergraphs whose edges contain 1 or 2 vertices in cite{PPTZ}. It is interesting if similar results hold for other non-uniform hypergraphs. In this paper, we give some connection between polynomial programming and the clique of non-uniform hypergraphs whose edges contain 1, or 2, and more vertices. Specifically, we obtain some Motzkin-Straus type results in terms of the graph-Lagrangian of non-uniform hypergraphs whose edges contain 1, or 2, and more vertices.
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It is conjectured by Frankl and Furedi that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$ edges in cite{FF}. Motzkin and Straus theorem confirms this conjecture when $r=2$. For $r=3$, it is shown by Talbot in cite{T} that this conjecture is true when $m$ is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for $r$-uniform hypergraphs. As an implication of this connection, we prove that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform graphs with $t$ vertices and $m$ edges satisfying ${t-1choose r}leq m leq {t-1choose r}+ {t-2choose r-1}-[(2r-6)times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2choose r-2}-1)$ for $rgeq 4.$
There is a remarkable connection between the maximum clique number and the Lagrangian of a graph given by T. S. Motzkin and E.G. Straus in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we explore evidences that the Lagrangian of a 3-uniform hypergraph is related to the order of its maximum cliques when the number of edges of the hypergraph is in certain range. In particular, we present some results about a conjecture introduced by Y. Peng and C. Zhao (2012) and describe a combinatorial algorithm that can be used to check the validity of the conjecture.
Frankl and Furedi conjectured in 1989 that the maximum Lagrangian, denoted by $lambda_r(m)$, among all $r$-uniform hypergraphs of fixed size $m$ is achieved by the minimum hypergraph $C_{r,m}$ under the colexicographic order. We say $m$ in {em principal domain} if there exists an integer $t$ such that ${t-1choose r}leq mleq {tchoose r}-{t-2choose r-2}$. If $m$ is in the principal domain, then Frankl-Furedis conjecture has a very simple expression: $$lambda_r(m)=frac{1}{(t-1)^r}{t-1choose r}.$$ Many previous results are focusing on $r=3$. For $rgeq 4$, Tyomkyn in 2017 proved that Frankl-F{u}redis conjecture holds whenever ${t-1choose r} leq m leq {tchoose r} -{t-2choose r-2}- delta_rt^{r-2}$ for a constant $delta_r>0$. In this paper, we improve Tyomkyns result by showing Frankl-F{u}redis conjecture holds whenever ${t-1choose r} leq m leq {tchoose r} -{t-2choose r-2}- delta_rt^{r-frac{7}{3}}$ for a constant $delta_r>0$.
Let $mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m times n$ incidence matrix of $mathcal{H}$ and let us denote $lambda =max_{v perp overline{1},|v| = 1}|Mv|$. We show that the discrepancy of $mathcal{H}$ is $O(sqrt{t} + lambda)$. As a corollary, this gives us that for every $t$, the discrepancy of a random $t$-regular hypergraph with $n$ vertices and $m geq n$ edges is almost surely $O(sqrt{t})$ as $n$ grows. The proof also gives a polynomial time algorithm that takes a hypergraph as input and outputs a coloring with the above guarantee.
If $G$ is a graph and $vec H$ is an oriented graph, we write $Gto vec H$ to say that every orientation of the edges of $G$ contains $vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We determine the threshold $p_{vec H}=p_{vec H}(n)$ for the property $G(n,p)to vec H$ for the cases in which $vec H$ is an acyclic orientation of a complete graph or of a cycle.
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