On Lagrangians of r-uniform hypergraphs

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in Can J Math 17:533–540 (1965). 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 Turán 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 Comb Probab Comput 11:199–216 (2002) and a result in Peng and Zhao (Graphs Comb, 29:681–694, 2013).

[1]  Yuejian Peng,et al.  On Finding Lagrangians of 3-uniform Hypergraphs , 2015, Ars Comb..

[2]  Marcello Pelillo,et al.  A Continuous Characterization of Maximal Cliques in k-Uniform Hypergraphs , 2008, LION.

[3]  Yuejian Peng,et al.  On Cliques and Lagrangians of 3-uniform Hypergraphs , 2012 .

[4]  Herbert S. Wilf,et al.  Spectral bounds for the clique and independence numbers of graphs , 1986, J. Comb. Theory, Ser. B.

[5]  Yuejian Peng,et al.  On Frankl and Füredi’s conjecture for 3-uniform hypergraphs , 2012, 1211.7056.

[6]  John M. Talbot.  Lagrangians Of Hypergraphs , 2002, Comb. Probab. Comput..

[7]  Marcello Pelillo,et al.  Generalizing the Motzkin-Straus Theorem to Edge-Weighted Graphs, with Applications to Image Segmentation , 2003, EMMCVPR.

[8]  Yuejian Peng,et al.  On Lagrangians of Hypergraphs and Cliques , 2012 .

[9]  Stanislav Busygin,et al.  A new trust region technique for the maximum weight clique problem , 2006, Discret. Appl. Math..

[10]  Vojtech Rödl,et al.  Hypergraphs do not jump , 1984, Comb..

[11]  Zoltán Füredi,et al.  Extremal problems whose solutions are the blowups of the small witt-designs , 1989, J. Comb. Theory, Ser. A.

[12]  Yuejian Peng,et al.  A Motzkin–Straus Type Result for 3-Uniform Hypergraphs , 2013, Graphs Comb..

[13]  Panos M. Pardalos,et al.  Continuous Characterizations of the Maximum Clique Problem , 1997, Math. Oper. Res..

[14]  Andrea Torsello,et al.  A Continuous-Based Approach for Partial Clique Enumeration , 2007, GbRPR.

[15]  P. Pardalos,et al.  A global optimization approach for solving the maximum clique problem , 1990 .

[16]  Dhruv Mubayi,et al.  A hypergraph extension of Turán's theorem , 2006, J. Comb. Theory, Ser. B.

[17]  Marco Budinich.  Exact Bounds on the Order of the Maximum Clique of a Graph , 2003, Discret. Appl. Math..

[18]  L. Moser,et al.  AN EXTREMAL PROBLEM IN GRAPH THEORY , 2001 .

[19]  T. Motzkin,et al.  Maxima for Graphs and a New Proof of a Theorem of Turán , 1965, Canadian Journal of Mathematics.

[20]  P. Erdös.  On an extremal problem in graph theory , 1970 .

[21]  Xiangde Zhang,et al.  Some results on Lagrangians of hypergraphs , 2014, Discret. Appl. Math..

[22]  Marcello Pelillo,et al.  A generalization of the Motzkin–Straus theorem to hypergraphs , 2009, Optim. Lett..