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On Graph-Lagrangians of Hypergraphs Containing Dense Subgraphs

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Abstract

Motzkin and Straus established a remarkable connection between the maximum clique and the Graph-Lagrangian of a graph 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 is useful in practice if similar results hold for hypergraphs. In this paper, we provide upper bounds on the Graph-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 Peng and Zhao (Graphs Comb. 29:681–694, 2013) and extend a result of Talbot (Comb. Probab. Comput. 11:199–216, 2002).

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References

  1. Turán, P.: On an extremal problem in graph theory. Mat. Fiz. Lapok 48, 436–452 (1941)

    MathSciNet  Google Scholar 

  2. Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math. 17, 533–540 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bomze, I.M.: Evolution towards the maximum clique. J. Glob. Optim. 10, 143–164 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Budinich, M.: Exact bounds on the order of the maximum clique of a graph. Discrete Appl. Math. 127, 535–543 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Busygin, S.: A new trust region technique for the maximum weight clique problem. Discrete Appl. Math. 154, 2080–2096 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gibbons, L.E., Hearn, D.W., Pardalos, P.M., Ramana, M.V.: Continuous characterizations of the maximum clique problem. Math. Oper. Res. 22, 754–768 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Pavan, M., Pelillo, M.: Generalizing the Motzkin–Straus theorem to edge-weighted graphs, with applications to image segmentation. In: Rangarajan, A., Figueiredo Mário, A.T., Zerubia, J. (eds.) Lecture Notes in Computer Science, vol. 2683, pp. 485–500. Spring, New York (2003)

    Google Scholar 

  8. Pardalos, P.M., Phillips, A.: A global optimization approach for solving the maximum clique problem. Int. J. Comput. Math. 33, 209–216 (1990)

    Article  MATH  Google Scholar 

  9. Buló, S.R., Torsello, A., Pelillo, M.: A continuous-based approach for partial clique enumeration. In: Escolano, F., Vento, M. (eds.) Lecture Notes in Computer Science, vol. 4538, pp. 61–70. Spring, New York (2007)

    Google Scholar 

  10. Sidorenko, A.F.: Solution of a problem of Bollobás on 4-graphs. Mat. Zametki 41, 433–455 (1987)

    MathSciNet  Google Scholar 

  11. Frankl, P., Füredi, Z.: Extremal problems whose solutions are the blow-ups of the small Witt-designs. J. Comb. Theory, Ser. A 52, 129–147 (1989)

    Article  MATH  Google Scholar 

  12. Frankl, P., Rödl, V.: Hypergraphs do not jump. Combinatorica 4, 149–159 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sós, V.T., Straus, E.G.: Extremals of functions on graphs with applications to graphs and hypergraphs. J. Comb. Theory, Ser. A 32, 246–257 (1982)

    Article  MATH  Google Scholar 

  14. Bulò, S.R., Pelillo, M.: A continuous characterization of maximal cliques in k-uniform hypergraphs. In: Maniezzo, V., Battiti, R., Watson, J.P. (eds.) Lecture Notes in Computer Science, vol. 5313, pp. 220–233. Spring, New York (2008)

    Google Scholar 

  15. Bulò, S.R., Pelillo, M.: A generalization of the Motzkin–Straus theorem to hypergraphs. Optim. Lett. 3, 287–295 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Peng, Y., Zhao, C.: A Motzkin–Straus type result for 3-uniform hypergraphs. Graphs Comb. 29, 681–694 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Talbot, J.M.: Lagrangians of hypergraphs. Comb. Probab. Comput. 11, 199–216 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mubayi, D.: A hypergraph extension of Turán’s theorem. J. Comb. Theory, Ser. B 96, 122–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Peng, Y., Tang, Q., Zhao, C.: On Lagrangians of r-uniform Hypergraphs. J. Comb. Optim. (2013). doi:10.1007/s10878-013-9671-3

    Google Scholar 

  20. Peng, Y., Zhu, H., Zheng, Y., Zhao, C.: On Cliques and Lagrangians of 3-uniform hypergraphs. Preprint (2012). arXiv:1211.6508

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Acknowledgements

We thank two anonymous referees and the editor for helpful and insightful comments.

We also thank Professor Franco Giannessi for suggesting the terminology ‘Graph-Lagrangian’.

This research is partially supported by National Natural Science Foundation of China (No. 11271116).

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Authors and Affiliations

  1. College of Sciences, Northeastern University, Shenyang, 110819, P.R. China

    Qingsong Tang & Xiangde Zhang

  2. School of Mathematics, Jilin University, Changchun, 130012, P.R. China

    Qingsong Tang & Cheng Zhao

  3. College of Mathematics, Hunan University, Changchun, 410082, P.R. China

    Yuejian Peng

  4. Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN, 47809, USA

    Cheng Zhao

Authors
  1. Qingsong Tang
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  2. Yuejian Peng
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  3. Xiangde Zhang
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  4. Cheng Zhao
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Corresponding author

Correspondence to Cheng Zhao.

Additional information

Communicated by Horst Martini.

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Tang, Q., Peng, Y., Zhang, X. et al. On Graph-Lagrangians of Hypergraphs Containing Dense Subgraphs. J Optim Theory Appl 163, 31–56 (2014). https://doi.org/10.1007/s10957-013-0485-3

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  • DOI: https://doi.org/10.1007/s10957-013-0485-3

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