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A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in [7]. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It has been also applied in spectral graph theory. Estimating the Lagrangians of hypergraphs has been successfully applied in the course of studying the Turan densities of several hypergraphs as well. It is useful in practice if Motzkin-Straus type results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus result to hypergraphs is false. We attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. In this paper, we give some Motzkin-Straus type results for r-uniform hypergraphs. These results generalize and refine a result of Talbot in [19] and a result in [11].
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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.
In 1965, Motzkin and Straus [5] provided a new proof of Turans theorem based on a continuous characterization of the clique number of a graph using the Lagrangian of a graph. This new proof aroused interests in the study of Lagrangians of r-uniform graphs. The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. Sidorenko and Frankl-Furedi applied Lagrangians of hypergraphs in finding Turan densities of hypergraphs. Frankl and Rodl applied it in disproving Erdos jumping constant conjecture. In most applications, we need an upper bound for the Lagrangian of a hypergraph. Frankl and Furedi conjectured that the r-uniform graph with m edges formed by taking the first m sets in the colex ordering of $N^(r)$ has the largest Lagrangian of all r-uniform graphs with m edges. Talbot in [14] provided some evidences for Frankl and Furedis conjecture. In this paper, we prove that the r-uniform graph with m edges formed by taking the first m sets in the colex ordering of $N^(r)$ has the largest Lagrangian of all r-uniform graphs on t vertices with m edges when ${t choose r}-3$ or ${t choose r}-4$. As an implication, we also prove that Frankl and Furedis conjecture holds for 3-uniform graphs with ${t choose 3}-3$ or ${t choose 3}-4$ edges.
Let $rge 3$. Given an $r$-graph $H$, the minimum codegree $delta_{r-1}(H)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(H)$ is contained in at least $t$ edges of $H$. Given an $r$-graph $F$, the codegree Turan density $gamma(F)$ is the smallest $gamma >0$ such that every $r$-graph on $n$ vertices with $delta_{r-1}(H)ge (gamma + o(1))n$ contains $F$ as a subhypergraph. Using results on the independence number of hypergraphs, we show that there are constants $c_1, c_2>0$ depending only on $r$ such that [ 1 - c_2 frac{ln t}{t^{r-1}} le gamma(K_t^r) le 1 - c_1 frac{ln t}{t^{r-1}}, ] where $K_t^r$ is the complete $r$-graph on $t$ vertices. This gives the best general bounds for $gamma(K_t^r)$.
Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph 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 provide upper bounds on the Lagrangian of a hypergraph containing dense subgraphs when the number of edges of the hypergraph is in certain ranges. These results support a pair of conjectures introduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002). keywords{Cliques of hypergraphs and Colex ordering and Lagrangians of hypergraphs and Polynomial optimization}
Frankl and Furedi (1989) conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${mathbb N}^{(r)}$ has the largest graph-Lagrangian of all $r$-graphs with $m$ edges. In this paper, we establish some bounds for graph-Lagrangians of some special $r$-graphs that support this conjecture.
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