Boundedness of optimal matrices in extremal multigraph and digraph problems

Abstract

We consider multigraphs in which any two vertices are joined by at mostq edges, and study the Turán-type problem for a given family of forbidden multigraphs. In the caseq=2, answering a question of Brown, Erdős and Simonovits, we obtain an explicit upper bound on the size of the matrix generating an asymptotical solution of the problem. In the caseq>2 we show that some analogous statements do not hold, and so disprove a conjecture of Brown, Erdős and Simonovits.

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References

  1. [1]

    W. G. Brown, P. Erdős, andM. Simonovits: Extremal problems for directed graphs,J. Combin. Theory B15 (1973), 77–93.

    Google Scholar 

  2. [2]

    W. G. Brown, P. Erdős, andM. Simonovits: Inverse extremal problems,Colloq. Math. Soc. János Bolyai 37, finity and Infinite Sets, Akad. Kiadó, Budapest, 1985, 119–156.

    Google Scholar 

  3. [3]

    W. G. Brown, P. Erdős, andM. Simonovits: Algorithmic solution of extremal digraph problemsTrans. of Amer. Math. Soc. 292 (1985) 421–449.

    Google Scholar 

  4. [4]

    P. Erdős, andL. Moser: On the representation of directed graphs as unions of orederings,Publ. Math. Inst. Hungar. Acad. Sci. 9 (1964), 125–132.

    Google Scholar 

  5. [5]

    P. Erdős, andM. Simonovits: A limit theorem in graph theory,Studia Sci. Math. Hungar. 1 (1966), 51–57.

    Google Scholar 

  6. [6]

    G. Katona, T. Nemetz, andM. Simonovits: On a graph problem of Turán, (in Hungarian),Mat. Lapok 15 (1964), 228–238.

    Google Scholar 

  7. [7]

    K. B. Reid, andE. T. Parker: Disproof of a conjecture of Erdős and Moser on tournaments,J. Combin. Theory 9 (1970), 225–238.

    Google Scholar 

  8. [8]

    I. J. Schoenberg: Metric spaces and positive definite functions,Trans. of Amer. Math. Soc. 44 (1938), 522–536.

    Google Scholar 

  9. [9]

    A. F. Sidorenko: On the maximal number of edges in a uniform hypergraph that does not contain prohibited subgraphs,Math. Notes 41 (1987), 247–259.

    Google Scholar 

  10. [10]

    A. F. Sidorenko: Inequalities in probability theory and Turán-type problems for graphs with colored vertices,Random Structures and Algorithms 2 (1992), 73–99.

    Google Scholar 

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Sidorenko, A. Boundedness of optimal matrices in extremal multigraph and digraph problems. Combinatorica 13, 109–120 (1993). https://doi.org/10.1007/BF01202793

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AMS subject classification code (1991)

  • 05 C 35
  • 05 C 50